123. Area of Circles and Parts of Circles

Learning Intentions

• Solve the area of a circle given its radius or diameter Use a calculator
• find the area of a circle given its radius or diameter using an approximation for
• To understand how to find the area of a semicircle or quadrant by multiplying a circle’s area by or

Pre-requisite Summary

• Know that area is the amount of surface inside a two-dimensional shape
• Know that area is measured in square units such as , and
• Know the meaning of radius and diameter in a circle
• Understand that the diameter is twice the radius
• Be able to Substitute a number into a formula
• Be able to Use a calculator to square a number
• Understand that an approximation is close to the exact value
• Know that a semicircle is half a circle and a quadrant is one quarter of a circle

Worked Examples

Worked Example 1

State the meaning of each term:

a) radius

b) diameter

c) area of a circle

Worked Example 2

Find the area of each circle using a calculator:

a) radius

b) diameter

c) radius

Worked Example 3

Find the area of each circle using :

a) radius

b) diameter

c) radius

Worked Example 4

A circle has diameter .

a) Find the radius

b) Find the area using a calculator

Worked Example 5

Find the area of each part of a circle using :

a) a semicircle with radius

b) a quadrant with radius

Worked Example 6

A circle has radius .

a) Find the area of the whole circle using a calculator

b) Find the area of the semicircle

c) Find the area of the quadrant

Problems

Problem 1

State the meaning of each term:

a) radius

b) diameter

c) area of a circle

Problem 2

Find the area of each circle using a calculator:

a) radius

b) diameter

c) radius

Problem 3

Find the area of each circle using :

a) radius

b) diameter

c) radius

Problem 4

A circle has diameter .

a) Find the radius

b) Find the area using a calculator

Problem 5

Find the area of each part of a circle using :

a) a semicircle with radius

b) a quadrant with radius

Problem 6

A circle has radius .

a) Find the area of the whole circle using a calculator

b) Find the area of the semicircle

c) Find the area of the quadrant

Exercises

Understanding and Fluency

Exercise 1.

Complete each statement:

a) The area of a circle is the amount of ______ inside the circle

b) The formula for the area of a circle is

c) The diameter is ______ times the radius

d) Area is measured in ______ units

Exercise 2.

Find the missing measure:

a) radius , diameter ?

b) diameter , radius ?

c) radius , diameter ?

d) diameter , radius ?

Exercise 3.

Find the area of each circle using a calculator:

a) radius

b) radius

c) diameter

Exercise 4.

Find the area of each circle using :

a) radius

b) diameter

c) radius

Exercise 5.

Find the area of each circle:

a) diameter , using a calculator

b) diameter , using

c) radius , using a calculator

Exercise 6.

Find the area of each part of a circle using :

a) a semicircle with radius

b) a quadrant with radius

c) a semicircle with diameter

Exercise 7.

Find the area of each part of a circle:

a) a quadrant with diameter

b) a semicircle with radius

c) a quadrant with radius

Exercise 8.

Solve each:

a) A circle has radius . Find the area of the whole circle and the semicircle using

b) A circle has diameter . Find the area of the whole circle and the quadrant using

c) A circle has radius . Find the area of the whole circle using a calculator

Reasoning

Exercise 9.

Explain why the formula for the area of a circle uses and not just .

Exercise 10.

A student says that if the diameter is , then the area is . Explain the mistake.

Exercise 11.

Noah says that a semicircle has area equal to . Is he correct? Explain.

Exercise 12.

Explain why the area of a quadrant is found by multiplying the area of the whole circle by .

Exercise 13.

A student finds the area of a circle and writes the answer in cm instead of . Describe the error.

Problem-solving

Exercise 14.

A circular garden has radius . Find its area using .

Exercise 15.

A clock face has diameter . Find its area using a calculator.

Exercise 16.

A semicircular window has radius . Find its area using .

Exercise 17.

A quadrant of a circle has radius . Find its area using .

Exercise 18.

A circular pond has diameter . Find the area of the pond using .

Exercise 19.

A pizza has radius . One quarter of the pizza is eaten. Find the area of the part that was eaten using .

Potential Misunderstandings

• Students may confuse area with circumference
• Students may confuse radius and diameter
• Students may forget to halve the diameter to find the radius before using the area formula
• Students may use instead of when calculating area.
• Students may forget to square the radius
• Students may square the diameter instead of the radius
• Students may round too early when using a calculator
• Students may forget that a semicircle is half of the whole circle
• Students may forget that a quadrant is one quarter of the whole circle
• Students may use linear units instead of square units in the final answer

Next: 124e. Sectors and Composite Areas