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Basic Geometry: Help & Review16 chapters | 109 lessons

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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.

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* = Ï€ *r*2

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?

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

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* = Ï€ (*d*/2)2

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

*A* = Ï€ (*d/2)2 *

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.

You realize that you can find the area if you know the diameter of your circular beach, so you go outside and measure the diameter of the circular area to find that it's 42 feet. You head back inside, and grab a piece of paper, a pencil, and a calculator to find the area (*A*) of the circle given that the diameter (*d*) is 42. Our first line states that:

*A* = Ï€(*d*/2)2

We plug in our known numbers in the second line:

*A* = Ï€(42/2)2

We calculate the division in line three:

*A* = Ï€(21)2

In line four, we find the square of 21:

*A* = Ï€(441)

And, finally, we arrive at the conclusion that ** A â‰ˆ 1385.44**.

We see that the area of your beach is going to be approximately 1385.44 square feet. Since the landscapers charge $0.50 per square foot, you are able to calculate the cost by multiplying 1385.44 â‹… 0.5.

**Cost = 1385.44 â‹… 0.5 = 692.72**

You calculate that creating the beach in your backyard will cost you $692.72. Wow! All that, just from knowing the diameter of the circle! Of course, this is just one example of how finding the area of a circle from its diameter can show up in our lives. You'll come across many more, so it's great that you now know exactly how to do this!

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. The **radius** of a circle is the length of the line segment from the center of the circle to any point on the circle. So a diameter is the same as two opposing radii. The **formula for the area of a circle** is *A* = Ï€ *r*2, where *r* is the length of the radius of a circle. We can use our knowledge that a diameter is made up of two radii to understand that *r* = *d*/2. With this knowledge, you can rewrite the formula for the area of a circle as *A* = Ï€ (*d*/2)2.

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Basic Geometry: Help & Review16 chapters | 109 lessons

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