GM Lesson 027 Areas of Circles and Sectors

Learning Intentions

By the end of this 45 minute lesson, students will be able to:

• Calculate the area of a circle using .
• Calculate the area of a sector using the central angle and radius.
• Interpret circular area calculations in practical contexts.

Prerequisites

Students should already be able to:

• Identify the radius and diameter of a circle.
• Use the relationship .
• Substitute values into a formula.
• Round answers to a given number of decimal places.
• Use on a calculator.
• Calculate areas of rectangles, triangles, parallelograms and trapeziums.

Key Idea Summary

The area of a circle is the amount of flat space inside the circle.

For a circle with radius ,

where:

• is the area of the circle
• is the radius

A sector is a part of a circle formed by two radii and an arc.

For a sector with central angle and radius ,

where:

• is the area of the sector
• is the central angle in degrees
• is the radius

A full circle has angle , so the fraction of the circle used is:

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Direct Instruction

A circle can be divided into many equal sectors. Since a full circle contains , the area of a sector is a fraction of the total area of the circle.

The fraction of the circle used is:

Therefore:

So:

Students should be careful to identify whether the given measurement is a radius or a diameter. If the diameter is given, first calculate the radius using:

Worked Example 1: Area of a Circle

Find the area of a circle with radius . Give your answer correct to decimal place.

Use:

Substitute :

Therefore, the area is:

Worked Example 2: Area of a Circle from Diameter

A circular table has diameter . Find the area of the tabletop correct to decimal places.

First find the radius:

Use:

Substitute :

Therefore, the area of the tabletop is:

Worked Example 3: Area of a Sector

Find the area of a sector with radius and central angle . Give your answer correct to decimal place.

Use:

Substitute and :

Therefore, the sector area is:

Worked Example 4: Practical Sector Area

A sprinkler waters a sector-shaped region of lawn. The sprinkler reaches and turns through an angle of . Find the area watered correct to decimal place.

Use:

Substitute and :

Therefore, the sprinkler waters approximately:

Understanding Checks

Check 1

A circle has radius .

Which formula should be used to calculate its area?

Check 2

A circle has diameter .

What is its radius?

Check 3

A sector has central angle .

What fraction of the full circle is this sector?

Check 4

A sector has central angle .

Explain why its area is half the area of the full circle.

Check 5

A sector has radius and central angle .

Write the correct substitution into the sector area formula, but do not calculate the final answer.

Exercises

Simple Familiar Exercises

Exercise 1

Find the area of a circle with radius . Give your answer correct to decimal place.

Exercise 2

Find the area of a circle with radius . Give your answer correct to decimal place.

Exercise 3

Find the area of a circle with diameter . Give your answer correct to decimal place.

Exercise 4

Find the area of a circle with diameter . Give your answer correct to decimal place.

Exercise 5

Find the area of a sector with radius and central angle . Give your answer correct to decimal place.

Exercise 6

Find the area of a sector with radius and central angle . Give your answer correct to decimal place.

Complex Familiar Exercises

Exercise 7

A circular garden bed has radius .

Calculate its area correct to decimal places.

Exercise 8

A circular sign has diameter .

Calculate the area of the sign correct to the nearest square centimetre.

Exercise 9

A pizza has diameter .

Calculate the area of the pizza correct to decimal place.

Exercise 10

A slice of cake is shaped like a sector. The cake has radius and the slice has central angle .

Calculate the area of the slice correct to decimal place.

Exercise 11

A rotating sprinkler waters a sector of radius and central angle .

Calculate the area watered correct to decimal place.

Exercise 12

A circular pond has diameter .

A quarter of the pond is covered with floating plants.

Calculate the area covered by the plants correct to decimal place.

Homework Problems

Problem 1

Find the area of a circle with radius . Give your answer correct to decimal place.

Problem 2

Find the area of a circle with diameter . Give your answer correct to decimal place.

Problem 3

Find the area of a sector with radius and central angle . Give your answer correct to decimal place.

Problem 4

Find the area of a sector with radius and central angle . Give your answer correct to decimal place.

Problem 5

A circular rug has diameter .

Calculate its area correct to decimal places.

Problem 6

A sprinkler waters a sector-shaped region with radius and central angle .

Calculate the area watered correct to decimal place.

Problem 7

A circular cake has radius . A slice has central angle .

Calculate the area of the slice correct to decimal place.

Problem 8

A circular garden has diameter . A quarter of the garden is planted with flowers.

Calculate the flower area correct to decimal place.

Problem 9

A circular metal plate has radius . A sector with angle is removed.

Calculate the area remaining correct to decimal place.

Problem 10

A circular playground has radius . Soft-fall rubber costs $ per square metre.

Calculate the total cost of covering the playground, correct to the nearest dollar.