Area and Circumference of Circles

Learning Objectives

Identify the radius and diameter of a circle.

Understand the meaning of the constant Pi (π).

Calculate the circumference of a circle using the formulas C = πd and C = 2πr.

Calculate the area of a circle using the formula A = πr².

The Geometry of Circles

A circle is a special shape defined as the set of all points that are the same distance from a central point. Circles are all around us, from the wheels on a car to the face of a clock. Understanding how to measure them is a key skill in geometry.

The Parts of a Circle

There are two fundamental measurements for a circle:

Radius (r): The distance from the center of the circle to any point on its edge.

Diameter (d): The distance across the circle passing through its center. The diameter is always exactly twice the length of the radius.

d = 2r

r = d / 2

The Magic Number: Pi (π)

There is a special relationship that is true for every circle, no matter how big or small. If you take the distance around a circle and divide it by the distance across it, you will always get the same number. This magical number is called Pi, and its symbol is the Greek letter π.

Pi is an irrational number, meaning its decimal representation goes on forever without repeating.

π ≈ 3.14159...

For most school calculations, we often use the approximation π ≈ 3.14 or the fraction π ≈ 22/7.

Circumference: The Distance Around

The circumference of a circle is the distance around its outer edge. It's like the perimeter of a polygon. Since Pi is the ratio of the circumference to the diameter (π = C/d), we can rearrange this to get the formula for circumference.

Formula 1 (using diameter): C = πd

Formula 2 (using radius): C = 2πr

Example: Find the circumference of a bicycle wheel with a diameter of 70 cm. (Use π ≈ 22/7)

Formula: C = πd

Substitute: C = (22/7) * 70 cm

Calculate: C = 22 (70/7) = 22 10 = 220 cm.

The circumference is 220 cm.

Area: The Space Inside

The area of a circle is the amount of two-dimensional space it covers.

Formula: A = πr²

Be careful! The formula uses the radius, not the diameter. If you are given the diameter, you must divide it by 2 first to find the radius before using the formula.

Example: Find the area of a pizza with a radius of 10 inches. (Use π ≈ 3.14)

Formula: A = πr²

Substitute: A = 3.14 * (10 inches)²

Calculate exponent first: A = 3.14 * (100 square inches)

Calculate: A = 314 square inches.

The area is 314 in².

Key Terms

**Circle: A round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center).
**Radius (r): The distance from the center of a circle to any point on its edge.
**Diameter (d): The distance across a circle passing through its center. (d = 2r).
**Pi (π): The irrational number (≈ 3.14) that represents the ratio of a circle's circumference to its diameter.
**Circumference (C): The distance around the outside of a circle.
**Area (A): The measure of the space inside a two-dimensional figure. The area of a circle is A = πr².