In this page, we will look at another measure of angles, that is, radians. This is a far more useful measure than degrees. Formula for calculating arcs and sectors become simpler and most importantly, we need to use radian measure whenever we are doing Calculus with trigonometric functions. Calculating arc lengths and sector areas is straightforward, but take care, some of the questions on this topic can be challenging. Make sure that you try some of the more difficult exam-style questions.
On this page, you should learn about
- radian measure for angles
- lengths of arcs
- areas of sectors
- areas of segments
Here is a quiz that practises converting degrees and radians
START QUIZ!
Here is a quiz that practises calculations with arc lengths, sector areas and segment areas
START QUIZ!
The following diagram shows a circle centre O and radius 5cm. A and B lie on the circumference of the circle and \(\large A\hat{O}B\) = 2 radians.
Find the length of the arc AB
length = cm
\(\large l=r\theta=5\times 2= 10\)
The following diagram shows a circle centre O and radius 5cm. A and B lie on the circumference of the circle and \(\large A\hat{O}B\) = 2 radians.
Find the area of the sector OAB
area = cm²
\(\large A=\frac{1}{2}r^2\theta=\frac{1}{2}\times5^2\times 2=25\)
The following diagram shows a circle centre O and radius 10cm. A and B lie on the circumference of the circle and \(\large A\hat{O}B\) = 80°.
The area of the sector OAB = \(\large \frac{a\pi}{9}\)cm²
Find the value of a
a =
\(\large A=\frac{\theta}{360}\pi r^2=\frac{80}{360}\pi \times10^2=\frac{200\pi}{9}\)
The following diagram shows a circle centre O and radius 10cm. A and B lie on the circumference of the circle and \(\large A\hat{O}B\) = 80°
The length of the arc AB =\(\large \frac{a\pi}{9}\)cm
Find the value of a
a =
\(\large l=\frac{\theta}{360}\times2\pi r=\frac{80}{360}\times2\pi \times10=\frac{40}{9}\pi\)
The following diagram shows a circle centre O and radius 4cm. A and B lie on the circumference of the circle and \(\large A\hat{O}B\) = 1 radian.
Find the area of the shaded region
The angle subtended at the centre of the circle is \(\large 2\pi -1\)
\(\large A=\frac{1}{2}r^2\theta=\frac{1}{2}\times4^2\times (2\pi-1)=8 (2\pi-1)\)
The following diagram shows a circle centre O and radius r. A and B lie on the circumference of the circle and \(\large A\hat{O}B\) = 60°.
The length of the arc is \(\large 4\pi\) cm.
Find the radius, r
r =
\(\large l=\frac{\theta}{360}\times 2\pi r\\ \large 4\pi=\frac{60}{360}\times 2\pi r\\ \large r=12\)
The following diagram shows a circle centre O and radius r. A and B lie on the circumference of the circle and \(\large A\hat{O}B=\theta\) radians.
The length of the arc AB = r.
Find \(\large \theta\)
\(\large \theta\) =
This is the definition of 1 radian.
The following diagram shows a circle centre O and radius 8cm. A and B lie on the circumference of the circle and \(\large A\hat{O}B=\theta\) radians.
The area of the sector OAB is 48cm².
Find the value of \(\large \theta\)
\(\large \theta\) =
\(\large A=\frac{1}{2}r^2\theta\\ \large 48=\frac{1}{2}\times 8^2\times\theta\\ \large \theta = \frac{48}{36}=1.5\)
The following diagram shows a circle centre O and radius 10cm. A and B lie on the circumference of the circle and \(\large A\hat{O}B=90°\)
Find the shaded area
Area of segment = area of sector - area of triangle
\(\large A=\frac{\pi \times10^2}{4}-\frac{10\times10}{2}\\ \large A=25\pi-50\)
The following diagram shows a circle centre O and radius r. A and B lie on the circumference of the circle and \(\large A\hat{O}B=\theta\) radians.
Find the shaded area
Area of segment = area of sector - area of triangle
\(\large A= \frac{1}{2}r^2\theta-\frac{1}{2}r^2\sin\theta\)
The following diagram shows a circle with centre O and radius 12cm. A and B lie on the circumference of the circle and \(\large AÔB=50°\)
a) Find the area of the minor sector OAB
b) Find the area of the triangle AOB
c) Hence, find the area of the shaded segment
Hint
Full Solution
The following diagram shows a circle with centre O and radius r cm
The area of the shaded sector OAB is \(\large \frac{40\pi}{3}\) cm²
The length of the minor arc AB is \(\large \frac{10\pi}{3}\) cm
a) Find the radius of the circle
b) Find the angle \(\large \theta\) , in radians
Hint
Full Solution
The following diagram shows a circle with centre O and radius 5cm and another circle with centre P and radius r. The two circles overlap meeting at points A and B. \(\large AÔP=45°\) and \(\large A\hat{P}O=30°\)
a) Show that \(\large r=5\sqrt{2}\) cm
b) Hence, show that the shaded area bounded by the two circles is \(\large \frac{25}{12}(7\pi-6-6\sqrt3)\) cm²
Hint
Full Solution
The following diagram shows a circle with centre O and radius r. A and B are points on the circumference of the circle and \(\large A\hat{O} B =\theta\) radians
The area of the green shaded region is three times greater than the area of the blue region.
a) Show that \(\large \sin \theta=\frac{4\theta-2\pi}{3}\)
b) Find the value of \(\large \theta\) , giving your answer correct to 3 significant figures.
Hint
Full Solution
The angle subtended by the the green sector is 
The following diagram shows a circle with centre O and radius r. Points A and B lie on the circumference of the circle and \(\large A\hat{O}B=\theta\) radians. The tangents to the circle A and B intersect at C.
a) Show that \(\large AC=r\tan (\frac{\theta}{2})\)
b) Hence, find the value of \(\large \theta\) when the two shaded regions have an equal area.
Hint
Full Solution
How much of Radians, Arcs and Sectors have you understood?
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