Topics
7. Geometry
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United States of America
Grade 7

7.03 Area of a circle

Lesson
Practice

Area of a circle

We already know that area is the space inside a 2D shape. We can find the area of a circle, but we will need a special rule.

Let's look at what happens when we unravel segments of a circle.

Exploration

  1. Slide the slider to unravel the circle. Explain how the width of the shape relates to the circumference of the circle.
  2. Move the triangle to slide the triangle together. What figure is formed? Explain how the area of this figure relates to the area of the circle.
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Interesting isn't it that when we realign the segments we end up with a parallelogram shape. Which is great, because it means we know how to find the area based on our knowledge that the area of a parallelogram has formula A=bh. In a circle, the base is half the circumference and the height is the radius.

\displaystyle \text{Area of a circle}\displaystyle =\displaystyle \dfrac{1}{2} \times 2\pi r \times r
\displaystyle =\displaystyle \pi r \times r
\displaystyle \text{Area of a circle}\displaystyle =\displaystyle \pi r^2

Examples

Example 1

Find the area of the circle shown, correct to one decimal place.

A circle with a radius of 6 centimeters.
Worked Solution
Create a strategy

The area of a circle can be found using the formula: A=\pi r^2.

Apply the idea
\displaystyle A\displaystyle =\displaystyle \pi \times (6)^2Substitute r=6
\displaystyle =\displaystyle 113.1 \text{ cm}^2Evaluate

Example 2

If the diameter of the circle is 24 cm, find its area correct to one decimal place.

Worked Solution
Create a strategy

Remember that the radius of a circle is half its diameter, r=\dfrac{d}{2}.

Apply the idea
\displaystyle r\displaystyle =\displaystyle \dfrac{24}{2}Divide the diameter by 2
\displaystyle =\displaystyle 12Evaluate
\displaystyle A\displaystyle =\displaystyle \pi \times 12^2Substitute r
\displaystyle =\displaystyle 452.4\text{ cm}^2Evaluate
Idea summary
\displaystyle \text{Area of a circle}=\pi r^2
\bm{r}
is the radius of the circle

Problem solving with areas of circles

Circles are everywhere. Using the formula for the area of the circle, we can find the area of different circular objects just by knowing their radius or diameter.

If finding the area of a circle uses a formula A=\pi r^2, we can also find dimensions of a circle such as radius and diameter given its area.

Examples

Example 3

Carlo and his friends ordered a pizza on a Saturday night. Each slice was 10\text{ cm} in length.

Find the area of the pizza that Carlo and his friends ordered. Use 3.14 to approximate \pi.

Worked Solution
Create a strategy

The length of the pizza is the radius of the pizza. use the formula of the area of the circle to find the area of the pizza.

Apply the idea
\displaystyle A\displaystyle =\displaystyle 3.14(10)^2Substitute the value of the \pi and the radius
\displaystyle =\displaystyle 314Evaluate

The area of the pizza is 314\text{ cm}^2.

Example 4

A circle has an area of 49\pi \text{ cm}^2.

a

What is its radius?

Worked Solution
Create a strategy

Rearrange the formula for the area of a circle to make r the subject of the equation. Substitute the values to find the radius.

Apply the idea
\displaystyle \pi r^2\displaystyle =\displaystyle 49\pi Substitute the value of the area to the equation
\displaystyle \dfrac{\pi r^2}{\pi}\displaystyle =\displaystyle \dfrac{49\pi}{\pi} Divide both sides of the equation by pi
\displaystyle r^2\displaystyle =\displaystyle 49 Evaluate
\displaystyle r \cdot r\displaystyle =\displaystyle 7 \cdot 7 Find the value of r that when multiplied to itself the answer is 49
\displaystyle r\displaystyle =\displaystyle 7

The radius is 7\text{ cm}.

b

What is its diameter?

Worked Solution
Create a strategy

The radius is always equal to half the diameter. This means that the diameter is always equal to twice the radius.

Apply the idea
\displaystyle \text{Diameter}\displaystyle =\displaystyle 2rThe equation for diameter given the radius of a circle
\displaystyle =\displaystyle 2(7)Substitute the value of r
\displaystyle =\displaystyle 14Evaluate

The diameter is 14 \text{ cm}.

Idea summary

Finding the area of a circle can be applied in daily life. We can find circles everywhere from clocks, loops, rings and even pizzas.

Given the area of a circle, we can can use the formula to find the radius and diameter.

Outcomes

7.G.B.4

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

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