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How to Find Area from Diameter

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  • 0:04 Diameter
  • 0:35 Radius
  • 1:09 Formula
  • 1:33 Application
  • 3:36 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will review what the diameter and the radius of a circle are and then look at how these two characteristics are related to one another. We will use this information to find a formula that enables us to find the area of a circle given its diameter.

Diameter

If you'd like to calculate the area of a circle from the diameter, you have to know what a diameter is. Let's take a look at what exactly is a diameter. The diameter of a circle is the length of the line segment that goes from one side of the circle to the other and passes through the center of the circle. Now, you may be familiar with the well-known formula for the area of a circle, which is:

A = π r2

where r is the length of the radius of a circle. But wait! What does this formula have to do with the diameter of the circle?

Radius

To figure this out, let's look at what a radius of a circle is. The radius of a circle is the length of the line segment from the center of the circle to any point on the circle.

Do you notice how the radius relates to the diameter? Since the diameter is the length of the line segment that goes through the center of the circle from one side of the circle to the other, it's actually made up of two radii. In other words, if d is the diameter of a circle and r is the radius of a circle, then d = 2r. We can also look at this as the radius is 1/2 of the diameter, or r = d / 2.

Formula

This is great news! Do you see why? All we have to do is plug r = d/2 into our area formula, and we have a way to find the area of a circle from its diameter.

A = π (d/2)2

Solution

If the length of the diameter of a circle is d, then we can find the area, A, using the following formula;

A = π (d/2)2

Application

Circles show up all the time in the world around us so, naturally, being able to find the area of a circle is extremely useful in real life. For example, suppose you want to create a sandy beach area in your backyard in such a way that the beach is circular in shape. You need to figure out how much sand will be required and how much it's going to cost. The landscapers you hired tell you that they determine how much sand you will need based on the area of the land that needs to be covered, and that they charge $0.50 per square foot.

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