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DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 3. Geometry & Trigonometry / 3.2 Geometry of 3D Shapes / 3.2.2 Volume & Surface Area


3.2.2 Volume & Surface Area


Volume of 3D Shapes

What is volume?

  • The volume of a 3D shape is a measure of how much 3D space it takes up
    • A 3D shape is also called a solid
  • You need to be able to calculate the volume of a number of common shapes

 

How do I find the volume of cuboids, prisms and cylinders?

  • A prism is a 3-D shape that has two identical base shapes connected by parallel edges
    • A prism has the same base shape all the way through
    • A prism takes its name from its base
  • To find the volume of any prism use the formula:

Volume of a prism = Ah

    • Where A is the area of the cross section and h is the base height
      • h could also be the length of the prism, depending on how it is oriented
    • This is in the formula booklet in the prior learning section at the beginning
    • The base could be any shape so as long as you know its area and length you can calculate the volume of any prism

Prism volume

  • Note two special cases:
    • To find the volume of a cuboid use the formula:

begin mathsize 22px style table attributes columnalign right center left columnspacing 0px end attributes row cell Volume space of space straight a space cuboid space end cell equals cell space length space cross times space width space cross times space height end cell row cell V space end cell equals cell space l w h end cell end table end style

    • The volume of a cylinder can be found in the same way as a prism using the formula:

begin mathsize 22px style Volume space of space straight a space cylinder space equals space pi italic space r to the power of italic 2 italic space h end style

    • where r is the radius, h is the height (or length, depending on the orientation
    • Note that a cylinder is technically not a prism as its base is not a polygon, however the method for finding its volume is the same
  • Both of these are in the formula booklet in the prior learning section

How do I find the volume of pyramids and cones?

  • In a right-pyramid the apex (the joining point of the triangular faces) is vertically above the centre of the base
    • The base can be any shape but is usually a square, rectangle or triangle
  • To calculate the volume of a right-pyramid use the formula

V equals 1 third A h

    • Where A is the area of the base, h is the height
    • Note that the height must be vertical to the base
  • A right cone is a circular-based pyramid with the vertical height joining the apex to the centre of the circular base
  • To calculate the volume of a right-cone use the formula

V equals 1 third pi space r to the power of 2 space end exponent h

    • Where r is the radius, h is the height
  • These formulae are both given in the formula booklet

 

How do I find the volume of a sphere?

  • To calculate the volume of a sphere use the formula

V equals 4 over 3 pi space r cubed

    • Where r is the radius
      • the line segment from the centre of the sphere to the surface
    • This formula is given in the formula booklet

Exam Tip

  • Remember to make use of the formula booklet in the exam as all the volume formulae you need will be here
    • Formulae for basic 3D objects (cuboid, cylinder and prism) are in the prior learning section
    • Formulae for other 3D objects (pyramid, cone and sphere) are in the Topic 3: Geometry section

Worked Example

A dessert can be modelled as a right-cone of radius 3 cm and height 12 cm and a scoop of ice-cream in the shape of a sphere of radius 3 cm.  Find the total volume of the ice-cream and cone.

diagram-for-we-3-2-2

3-2-2-ai-sl-volume-we-solution

Surface Area of 3D Shapes

What is surface area?

  • The surface area of a 3D shape is the sum of the areas of all the faces that make up a shape
    • A face is one of the flat or curved surfaces that make up a 3D shape
    • It often helps to consider a 3D shape in the form of its 2D net

 

How do I find the surface area of cuboids, pyramids and prisms?

  • Any prisms and pyramids that have polygons as their bases have only flat faces
    • The surface area is simply found by adding up the areas of these flat faces
    • Drawing a 2D net will help to see which faces the 3D shape is made up of

How do I find the surface area of cylinders, cones and spheres?

  • Cones, cylinders and spheres all have curved faces so it is not always as easy to see their shape
    • The net of a cylinder is made up of two identical circles and a rectangle
    • The rectangle is the curved surface area and is harder to identify
    • The length of the rectangle is the same as the circumference of the circle
    • The area of the curved surface area is

begin mathsize 22px style A equals 2 pi r h end style

      • where r is the radius, h is the height
    • This is given in the formula book in the prior learning section
    • The area of the total surface area of a cylinder is

begin mathsize 22px style A equals 2 pi r h plus 2 pi r squared blank end style

    • This is not given in the formula book, however it is easy to put together as both the area of a circle and the area of the curved surface area are given
  • The net of a cone consists of the circular base along with the curved surface area
    • The area of the curved surface area is

begin mathsize 22px style A blank equals pi r l end style

      • Where r is the radius and l is the slant height
    • This is given in the formula book
      • Be careful not to confuse the slant height, l, with the vertical height, h
      • Note that r, h and l will create a right-triangle with l as the hypotenuse
    • The area of the total surface area of a cone is

begin mathsize 22px style A equals pi r l plus pi r squared blank end style

    • This is not given in the formula book, however it is easy to put together as both the area of a circle and the area of the curved surface area are given
  • To find the surface area of a sphere use the formula

A equals 4 pi r squared

      • where r is the radius (line segment from the centre to the surface)
    • This is given in the formula booklet, you do not have to remember it

Exam Tip

  • Remember to make use of the formula booklet in the exam as all the area formulae you need will be here
    • Formulae for basic 2D shapes (parallelogram, triangle, trapezoid, circle, curved surface of a cylinder) are in the prior learning section
    • Formulae for other 2D shapes (curved surface area of a cone and surface area of a sphere ) are in the Topic 3: Geometry section

Worked Example

In the diagram below ABCD  is the square base of a right pyramid with vertex V .  The centre of the base is M. The sides of the square base are 3.6 cm and the vertical height is 8.2 cm.

sa-diagram-for-we-3-2-2

i)
Use the Pythagorean Theorem to find the distance VN.

 3-2-2-ai-sl-surface-area-we-solution-i

ii)
Calculate the area of the triangle ABV.

 3-2-2-ai-sl-surface-area-we-solution-ii

iii)
Find the surface area of the right pyramid.

3-2-2-ai-sl-surface-area-we-solution-iii



  • 1. Number & Algebra
    • 1.1 Number & Algebra Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
          • 1.1.3 Partial Fractions
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              • 1.2.2 Laws of Logarithms
                • 1.2.3 Solving Exponential Equations
                • 1.3 Sequences & Series
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                      • 1.3.3 Geometric Sequences & Series
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                            • 1.5 Further Proof & Reasoning
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                                • 1.6 Binomial Theorem
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                                    • 1.7 Permutations & Combinations
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                                        • 1.8 Complex Numbers
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                                            • 1.8.2 Modulus & Argument
                                              • 1.8.3 Introduction to Argand Diagrams
                                              • 1.9 Further Complex Numbers
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                                                  • 1.9.2 Forms of Complex Numbers
                                                    • 1.9.3 Complex Roots of Polynomials
                                                      • 1.9.4 De Moivre's Theorem
                                                        • 1.9.5 Roots of Complex Numbers
                                                        • 1.10 Systems of Linear Equations
                                                          • 1.10.1 Systems of Linear Equations
                                                            • 1.10.2 Algebraic Solutions
                                                          • 2. Functions
                                                            • 2.1 Linear Functions & Graphs
                                                              • 2.1.1 Equations of a Straight Line
                                                              • 2.2 Quadratic Functions & Graphs
                                                                • 2.2.1 Quadratic Functions
                                                                  • 2.2.2 Factorising & Completing the Square
                                                                    • 2.2.3 Solving Quadratics
                                                                      • 2.2.4 Quadratic Inequalities
                                                                        • 2.2.5 Discriminants
                                                                        • 2.3 Functions Toolkit
                                                                          • 2.3.1 Language of Functions
                                                                            • 2.3.2 Composite & Inverse Functions
                                                                              • 2.3.3 Symmetry of Functions
                                                                                • 2.3.4 Graphing Functions
                                                                                • 2.4 Other Functions & Graphs
                                                                                  • 2.4.1 Exponential & Logarithmic Functions
                                                                                    • 2.4.2 Solving Equations
                                                                                      • 2.4.3 Modelling with Functions
                                                                                      • 2.5 Reciprocal & Rational Functions
                                                                                        • 2.5.1 Reciprocal & Rational Functions
                                                                                        • 2.6 Transformations of Graphs
                                                                                          • 2.6.1 Translations of Graphs
                                                                                            • 2.6.2 Reflections of Graphs
                                                                                              • 2.6.3 Stretches Graphs
                                                                                                • 2.6.4 Composite Transformations of Graphs
                                                                                                • 2.7 Polynomial Functions
                                                                                                  • 2.7.1 Factor & Remainder Theorem
                                                                                                    • 2.7.2 Polynomial Division
                                                                                                      • 2.7.3 Polynomial Functions
                                                                                                        • 2.7.4 Roots of Polynomials
                                                                                                        • 2.8 Inequalities
                                                                                                          • 2.8.1 Solving Inequalities Graphically
                                                                                                            • 2.8.2 Polynomial Inequalities
                                                                                                            • 2.9 Further Functions & Graphs
                                                                                                              • 2.9.1 Modulus Functions
                                                                                                                • 2.9.2 Modulus Transformations
                                                                                                                  • 2.9.3 Modulus Equations & Inequalities
                                                                                                                    • 2.9.4 Reciprocal & Square Transformations
                                                                                                                  • 3. Geometry & Trigonometry
                                                                                                                    • 3.1 Geometry Toolkit
                                                                                                                      • 3.1.1 Coordinate Geometry
                                                                                                                        • 3.1.2 Radian Measure
                                                                                                                          • 3.1.3 Arcs & Sectors
                                                                                                                          • 3.2 Geometry of 3D Shapes
                                                                                                                            • 3.2.1 3D Coordinate Geometry
                                                                                                                              • 3.2.2 Volume & Surface Area
                                                                                                                              • 3.3 Trigonometry Toolkit
                                                                                                                                • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                                                  • 3.3.2 Non Right-Angled Trigonometry
                                                                                                                                    • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                                                    • 3.4 Trigonometry
                                                                                                                                      • 3.4.1 The Unit Circle
                                                                                                                                        • 3.4.2 Exact Values
                                                                                                                                        • 3.5 Trigonometric Functions & Graphs
                                                                                                                                          • 3.5.1 Graphs of Trigonometric Functions
                                                                                                                                            • 3.5.2 Transformations of Trigonometric Functions
                                                                                                                                              • 3.5.3 Modelling with Trigonometric Functions
                                                                                                                                              • 3.6 Trigonometric Equations & Identities
                                                                                                                                                • 3.6.1 Simple Identities
                                                                                                                                                  • 3.6.2 Compound Angle Formulae
                                                                                                                                                    • 3.6.3 Double Angle Formulae
                                                                                                                                                      • 3.6.4 Relationship Between Trigonometric Ratios
                                                                                                                                                        • 3.6.5 Linear Trigonometric Equations
                                                                                                                                                          • 3.6.6 Quadratic Trigonometric Equations
                                                                                                                                                          • 3.7 Inverse & Reciprocal Trig Functions
                                                                                                                                                            • 3.7.1 Reciprocal Trig Functions
                                                                                                                                                              • 3.7.2 Inverse Trig Functions
                                                                                                                                                              • 3.8 Further Trigonometry
                                                                                                                                                                • 3.8.1 Trigonometric Proof
                                                                                                                                                                  • 3.8.2 Strategy for Trigonometric Equations
                                                                                                                                                                  • 3.9 Vector Properties
                                                                                                                                                                    • 3.9.1 Introduction to Vectors
                                                                                                                                                                      • 3.9.2 Position & Displacement Vectors
                                                                                                                                                                        • 3.9.3 Magnitude of a Vector
                                                                                                                                                                          • 3.9.4 The Scalar Product
                                                                                                                                                                            • 3.9.5 Geometric Proof with Vectors
                                                                                                                                                                            • 3.10 Vector Equations of Lines
                                                                                                                                                                              • 3.10.1 Vector Equations of Lines
                                                                                                                                                                                • 3.10.2 Applications to Kinematics
                                                                                                                                                                                  • 3.10.3 Pairs of Lines in 3D
                                                                                                                                                                                    • 3.10.4 The Vector Product
                                                                                                                                                                                      • 3.10.5 Shortest Distances with Lines
                                                                                                                                                                                      • 3.11 Vector Planes
                                                                                                                                                                                        • 3.11.1 Vector Equations of Planes
                                                                                                                                                                                          • 3.11.2 Intersections of Lines & Planes
                                                                                                                                                                                            • 3.11.3 Angles Between Lines & Planes
                                                                                                                                                                                              • 3.11.4 Shortest Distances with Planes
                                                                                                                                                                                            • 4. Statistics & Probability
                                                                                                                                                                                              • 4.1 Statistics Toolkit
                                                                                                                                                                                                • 4.1.1 Sampling & Data Collection
                                                                                                                                                                                                  • 4.1.2 Statistical Measures
                                                                                                                                                                                                    • 4.1.3 Frequency Tables
                                                                                                                                                                                                      • 4.1.4 Linear Transformations of Data
                                                                                                                                                                                                        • 4.1.5 Outliers
                                                                                                                                                                                                          • 4.1.6 Univariate Data
                                                                                                                                                                                                            • 4.1.7 Interpreting Data
                                                                                                                                                                                                            • 4.2 Correlation & Regression
                                                                                                                                                                                                              • 4.2.1 Bivariate Data
                                                                                                                                                                                                                • 4.2.2 Correlation & Regression
                                                                                                                                                                                                                • 4.3 Probability
                                                                                                                                                                                                                  • 4.3.1 Probability & Types of Events
                                                                                                                                                                                                                    • 4.3.2 Conditional Probability
                                                                                                                                                                                                                      • 4.3.3 Bayes' Theorem
                                                                                                                                                                                                                        • 4.3.4 Sample Space Diagrams
                                                                                                                                                                                                                        • 4.4 Probability Distributions
                                                                                                                                                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                                                                                                                                                            • 4.4.2 Mean & Variance
                                                                                                                                                                                                                            • 4.5 Binomial Distribution
                                                                                                                                                                                                                              • 4.5.1 The Binomial Distribution
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                                                                                                                                                                                                                                • 4.6 Normal Distribution
                                                                                                                                                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                                                                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                                                                                                                                                      • 4.6.3 Standardisation of Normal Variables
                                                                                                                                                                                                                                      • 4.7 Further Probability Distributions
                                                                                                                                                                                                                                        • 4.7.1 Probability Density Function
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                  • 5.2.3 Higher Order Derivatives
                                                                                                                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                                                                                                                        • 5.3 Integration
                                                                                                                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                          • 5.7 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.7.1 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.8 Advanced Differentiation
                                                                                                                                                                                                                                                                              • 5.8.1 First Principles Differentiation
                                                                                                                                                                                                                                                                                • 5.8.2 Applications of Chain Rule
                                                                                                                                                                                                                                                                                  • 5.8.3 Implicit Differentiation
                                                                                                                                                                                                                                                                                    • 5.8.4 Differentiating Further Functions
                                                                                                                                                                                                                                                                                    • 5.9 Advanced Integration
                                                                                                                                                                                                                                                                                      • 5.9.1 Integrating Further Functions
                                                                                                                                                                                                                                                                                        • 5.9.2 Further Techniques of Integration
                                                                                                                                                                                                                                                                                          • 5.9.3 Integrating with Partial Fractions
                                                                                                                                                                                                                                                                                            • 5.9.4 Advanced Applications of Integration
                                                                                                                                                                                                                                                                                              • 5.9.5 Modelling with Volumes of Revolution
                                                                                                                                                                                                                                                                                              • 5.10 Differential Equations
                                                                                                                                                                                                                                                                                                • 5.10.1 Numerical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                  • 5.10.2 Analytical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.10.3 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.11 MacLaurin Series
                                                                                                                                                                                                                                                                                                      • 5.11.1 Maclaurin Series
                                                                                                                                                                                                                                                                                                        • 5.11.2 Maclaurin Series from Differential Equations
                                                                                                                                                                                                                                                                                                        • 5.12 Further Limits (inc l'Hôpital's Rule)
                                                                                                                                                                                                                                                                                                          • 5.12.1 Further Limits
                                                                                                                                                                                                                                                                                                        Amber Perry

                                                                                                                                                                                                                                                                                                        Author: Amber

                                                                                                                                                                                                                                                                                                        Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.


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