Circles- Area and Circumference

Circles Are Everywhere

When was the last time you jumped on a trampoline? Did you ever wonder how much material was required for the center mat or how much steel was needed to create the frame? If you did, you may not have realized that both the mat and the frame are examples of how area and circumference of circles are used in real life. Let’s take a brief look at each of these concepts and examine how they are used in both two-dimensional and three-dimensional situations.

Area

As you may recall, area is the amount of space taken up by a two-dimensional figure, and it is measured in units squared. Most shapes require a formula to calculate area, and circles are no exception. To calculate the area of a circle, use the formula A = pi r ^2. In the formula, r represents the radius , which is a segment that connects the center of the circle to a point on the edge of the circle. It is half the size of the diameter, and every radius inside of a circle will be the same size. Let’s take a brief look at how to calculate area of circles.

Take a look at circle ‘m’. Here, we see that the diameter is 12 inches long. To calculate the area of circle ‘m,’ we must cut the diameter in half to determine the radius. When we divide 12 by 2, we see that the radius has a length of 6 inches. From here, we will substitute 6 into the equation to get A = pi (6)^2. Remember to follow the order of operations - we must square the radius before multiplying by pi. When we do, we are able to determine that the total area of circle ‘m’ is 36 pi or 113.097 inches squared.

Now, let’s take a look at a real-world example. Joe is purchasing material to build his first trampoline. If he wants the diameter of the mat to be 14 feet long, how much nylon will he have to purchase?

Once again, we will cut the diameter in half to determine the radius. When we do, we will see that the radius of the mat will be 7 feet. Substituting this radius into the equation gives us A = pi (7)^2. Once we complete our calculations using the order of operations, we will find that the total area of the mat is 49 pi or 153.938 feet squared.

Circumference

So, that takes care of calculating the amount of space taken up by a circle, but how can we determine the total distance around a circle? For other shapes, this is referred to as perimeter, but circles don’t have actual sides like other shapes. Therefore, the term circumference is used. It is defined as the distance around a circle and is represented by the formula C = 2 pi r , where r is the radius.

Let’s examine a few problems, beginning with Joe’s trampoline. In addition to buying nylon for the mat, Joe needs to create a steel frame. With a diameter of 14 feet, how much steel must he purchase to build a frame that will enclose the entire mat?

As you recall, with a diameter of 14 feet, the radius will be 7 feet. Let’s plug this into the formula for circumference. Once we do, we get 2 pi (7). After simplifying, we see that the circumference of the mat is 14 pi or 43.98 feet, and that is how much steel Joe will need to purchase for the frame.

Here’s one more. Clara has started her own hat decorating business. For her current design, she wants to line the brim of the hat with ribbon. If the brim has a diameter of 24 inches, how much ribbon will she need?

Since the diameter of the hat is 24 inches, we must divide this number by 2 to determine the radius. When we do, we see that the radius is 12 inches and we are ready to plug this into the formula. Doing so gives us 2 pi (12), which will equal 24 pi or 75.398 inches. Therefore, she will need 75.398 inches of ribbon to line her hat.

Circles in 3D

In addition to these examples, it’s important to realize that the formulas for area and circumference of circles create the foundation for calculating the surface area and volume of some three-dimensional figures, specifically cylinders, cones and spheres.

Recall that surface area is the amount of area taken up by each side of a figure, and volume is the amount of space inside of a figure, or the amount that a three-dimensional figure can hold. Let’s look at each, beginning with cylinders.

The surface area of a cylinder is calculated with the formula SA = (2 pi r ^2) + (2 pi r h ). In the first part of the formula, you can see the area of a circle. It is being multiplied by 2, because there are two circles that form the surface of the cylinder. In the second part of the formula, we see the circumference. It is being multiplied by the height of the cylinder to calculate the area for the rounded side of the figure.

Cylinder volume is calculated with the formula V = pi r ^2 h . The area of a circle shows up here as well. This time, it’s being multiplied by the height of the cylinder to determine the amount needed to fill the entire figure.

Now let’s take a look at the role they play with surface area and volume of cones. Can you spot the area or circumference formula for circles?

SA = (pi r l ) + (pi r ^2)

V = (1/3) pi r ^2 h

In both of these formulas, we can see that the formula for the area of a circle is required.

Now, let’s take a look at surface area for spheres. What formula do you see?

SA = 4 pi r ^2

Here, once again, we have the area of a circle serving as the foundation for this formula.

Without the ability to calculate the area and circumference of a circle, finding surface area and volume of cylinders, cones and spheres would not be possible.

Lesson Summary

In review, the area of a circle is defined as the amount of space covered by the circle and is calculated using the formula A = pi r ^2. Circumference , the distance around the circle, is calculated using the formula C = 2 pi r . For both formulas, r represents the radius of the circle.

From jumping on a trampoline to designing hats or baking cakes, area and circumference of circles are present in everyday life, and they form the foundation to calculate volume and surface area of cylinders, cones and spheres.