Radians, Arcs and Sectors

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.


Key Concepts

On this page, you should learn about

  • radian measure for angles
  • lengths of arcs
  • areas of sectors
  • areas of segments

Summary

Test Yourself

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!

Exam-style Questions

Question 1

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

 

Question 2

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

Question 3

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

 

Question 4

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

Question 5

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

MY PROGRESS

How much of Radians, Arcs and Sectors have you understood?