Circles: Area

Study Guide

Key Definition

The area of a circle is the number of square units that can fit inside that circle. It's given by the formula $A = \pi r^2$ where <a href="radius">$r$</a> is the radius of the circle.

Important Notes

The radius of a circle is half its diameter.
The constant <a href="pi">$\pi$</a> is approximately equal to 3.14
The area of a circle increases quadratically with its radius.
Units of area are always square units (such as square inches, square feet, or square meters).

Mathematical Notation

<a href="pi">$\pi$</a> is the mathematical constant approximately equal to 3.14

<a href="radius">$r$</a> is the radius of the circle

<a href="diameter">$d$</a> is the diameter of the circle, $d = 2r$

Remember to use proper notation when solving problems

Why It Works

To derive $A = \pi r^2$, imagine dividing the circle into n equal sectors each of central angle 2π/n. Each sector can be viewed as a narrow triangle whose base is the arc length (2πr/n) and whose height is the radius r. If you rearrange these sectors alternately, they form a shape that approaches a parallelogram with base equal to half the circumference (πr) and height r. Taking the limit as n → ∞ makes this exact, giving area = base × height = πr · r = πr².

Remember

Always be careful with units when calculating the area. If the radius is given in feet, the area will be in square feet.

Quick Reference

Area of a circle:$A = \pi r^2$

Understanding Circles: Area

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Video explanation of this concept

Beginner

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Beginner Explanation

The area of a circle is the space enclosed by the circle. It is calculated using the formula $A = \pi r^2$ where <a href="radius">$r$</a> is the radius of the circle.

Practice Problems

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Quick Quiz

Single Choice Quiz

Beginner

What is the area of a circle with radius $4 cm$?

Real-World Problem

Question Exercise

Intermediate

Pizza Party

Your family is having a pizza party and you're in charge of ordering the pizzas. The pizza place offers two sizes: small with a diameter of 10 inches and large with a diameter of 16 inches. Which size gives more pizza per inch of diameter?

Advanced Challenge

Thinking Exercise

Advanced

Think About This

If the radius of a circle is doubled, what happens to its area?

Recap

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