Learn on PengiPengi Math (Grade 7)Chapter 7: 2D Geometry and Measurement

Lesson 5: Area of Circles

In this Grade 7 Pengi Math lesson from Chapter 7: 2D Geometry and Measurement, students derive and apply the formula for the area of a circle, practicing conversion between diameter and radius to find exact and approximate areas. The lesson extends to calculating the area of semicircles and quadrants before applying these skills to real-world problems involving circular area.

Section 1

Area of a circle

Property

The formula for area of a circle is:

A = \pi r^2

Examples

A circular sandbox has a radius of 2.5 feet. Its area is $A = \pi r^2 \approx 3.14(2.5)^2 = 3.14(6.25) = 19.625$ square feet.

The lid of a paint bucket has a radius of 7 inches. Using $\pi \approx \frac{22}{7}$ , the area is $A \approx \frac{22}{7}(7)^2 = \frac{22}{7}(49) = 154$ square inches.

Section 2

Area of a Semicircle

Property

A semicircle is exactly half of a full circle. To find its area, calculate the area of the full circle first, and then simply divide by 2:

A = \frac{\pi r^2}{2}

Examples

Find the area of a semicircle with a radius of 6 cm. Full circle area is $36\pi$ . Semicircle area is $36\pi / 2 = 18\pi$ square cm (about 56.52 square cm).
A semicircular window has a diameter of 8 feet. First, find the radius (4 feet). Full circle area is $16\pi$ . Semicircle area is $16\pi / 2 = 8\pi$ square feet.

Explanation

Don't let semicircles trick you! There is no need to memorize a completely new, complicated formula. Just pretend it is a perfectly normal, full circle, do your standard $\pi r^2$ math, and at the very end, chop your answer in half.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Area of a circle

Property

The formula for area of a circle is:

A = \pi r^2

Examples

A circular sandbox has a radius of 2.5 feet. Its area is $A = \pi r^2 \approx 3.14(2.5)^2 = 3.14(6.25) = 19.625$ square feet.

The lid of a paint bucket has a radius of 7 inches. Using $\pi \approx \frac{22}{7}$ , the area is $A \approx \frac{22}{7}(7)^2 = \frac{22}{7}(49) = 154$ square inches.

Section 2

Area of a Semicircle

Property

A semicircle is exactly half of a full circle. To find its area, calculate the area of the full circle first, and then simply divide by 2:

A = \frac{\pi r^2}{2}

Examples

Find the area of a semicircle with a radius of 6 cm. Full circle area is $36\pi$ . Semicircle area is $36\pi / 2 = 18\pi$ square cm (about 56.52 square cm).
A semicircular window has a diameter of 8 feet. First, find the radius (4 feet). Full circle area is $16\pi$ . Semicircle area is $16\pi / 2 = 8\pi$ square feet.

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Area of a circle

Property

The formula for area of a circle is:

A = \pi r^2

Examples

A circular sandbox has a radius of 2.5 feet. Its area is $A = \pi r^2 \approx 3.14(2.5)^2 = 3.14(6.25) = 19.625$ square feet.

The lid of a paint bucket has a radius of 7 inches. Using $\pi \approx \frac{22}{7}$ , the area is $A \approx \frac{22}{7}(7)^2 = \frac{22}{7}(49) = 154$ square inches.

Section 2

Area of a Semicircle

Property

A semicircle is exactly half of a full circle. To find its area, calculate the area of the full circle first, and then simply divide by 2:

A = \frac{\pi r^2}{2}

Examples

Find the area of a semicircle with a radius of 6 cm. Full circle area is $36\pi$ . Semicircle area is $36\pi / 2 = 18\pi$ square cm (about 56.52 square cm).
A semicircular window has a diameter of 8 feet. First, find the radius (4 feet). Full circle area is $16\pi$ . Semicircle area is $16\pi / 2 = 8\pi$ square feet.