Circles

 

Learning Objective(s)

·         Identify properties of circles.

·         Find the circumference of a circle.

·         Find the area of a circle.

·         Find the area and perimeter of composite geometric figures.

 

 

Circles are a common shape. You see them all over—wheels on a car, Frisbees passing through the air, compact discs delivering data. These are all circles.

 

A circle is a two-dimensional figure just like polygons and quadrilaterals. However, circles are measured differently than these other shapes—you even have to use some different terms to describe them. Let’s take a look at this interesting shape.

 

 

A circle represents a set of points, all of which are the same distance away from a fixed, middle point. This fixed point is called the center. The distance from the center of the circle to any point on the circle is called the radius.

 

When two radii (the plural of radius) are put together to form a line segment across the circle, you have a diameter. The diameter of a circle passes through the center of the circle and has its endpoints on the circle itself.

 

 

The diameter of any circle is two times the length of that circle’s radius. It can be represented by the expression 2r, or “two times the radius.” So if you know a circle’s radius, you can multiply it by 2 to find the diameter; this also means that if you know a circle’s diameter, you can divide by 2 to find the radius.

 

 

 

 

 

 

 

The distance around a circle is called the circumference. (Recall, the distance around a polygon is the perimeter.)

 

One interesting property about circles is that the ratio of a circle’s circumference and its diameter is the same for all circles. No matter the size of the circle, the ratio of the circumference and diameter will be the same.

 

Some actual measurements of different items are provided below. The measurements are accurate to the nearest millimeter or quarter inch (depending on the unit of measurement used). Look at the ratio of the circumference to the diameter for each one—although the items are different, the ratio for each is approximately the same.

 

 

The circumference and the diameter are approximate measurements, since there is no precise way to measure these dimensions exactly. If you were able to measure them more precisely, however, you would find that the ratio  would move towards 3.14 for each of the items given. The mathematical name for the ratio  is pi, and is represented by the Greek letter .

 

 is a non-terminating, non-repeating decimal, so it is impossible to write it out completely. The first 10 digits of  are 3.141592653; it is often rounded to 3.14 or estimated as the fraction . Note that both 3.14 and  are approximations of, and are used in calculations where it is not important to be precise.

 

Since you know that the ratio of circumference to diameter (or ) is consistent for all circles, you can use this number to find the circumference of a circle if you know its diameter.

 

 = , so C = d

 

Also, since d = 2r, then C = d = (2r) = 2r.

 

 

Circumference of a Circle   To find the circumference (C) of a circle, use one of the following formulas:   If you know the diameter (d) of a circle: If you know the radius (r) of a circle:

 

 

 

 

 

 

A circle has a radius of 8 inches. What is its circumference, rounded to the nearest inch?   A) 25 inches B) 50 inches C) 64 inches2 D) 201 inches   Show/Hide Answer A) 25 inches Incorrect. You multiplied the radius times ; the correct formula for circumference when the radius is given is The correct answer is 50 inches.   B) 50 inches Correct. If the radius is 8 inches, the correct formula for circumference when the radius is given is The correct answer is 50 inches.   C) 64 inches2 Incorrect. You squared 8 inches to arrive at the answer 64 inches2; this will give you the area of a square with sides of 8 inches. Remember that the formula for circumference when the radius is given is . The correct answer is 50 inches.   D) 201 inches Incorrect. It looks like you squared 8 and then multiplied 64 by  to arrive at this answer. Remember that the formula for circumference when the radius is given is . The correct answer is 50 inches.

 

 

 

 is an important number in geometry. You have already used it to calculate the circumference of a circle. You use  when you are figuring out the area of a circle, too.

 

Area of a Circle   To find the area (A) of a circle, use the formula:

 

 

 

 

A button has a diameter of 20 millimeters. What is the area of the button? Use 3.14 as an approximation of .   A) 62.8 mm B) 314 mm2 C) 400 mm2 D) 1256 mm2   Show/Hide Answer A) 62.8 mm Incorrect. You found the circumference of the button: 20 • 3.14 = 62.8. To find the area, use the formula . The correct answer is 314 mm2.   B) 314 mm2 Correct. The diameter is 20 mm, so the radius must be 10 mm. Then, using the formula , you find  mm2.   C) 400 mm2 Incorrect. You squared 20 to arrive at 400 mm2; this gives you the area of a square with sides of length 20, not the area of a circle. To find the area, use the formula . The correct answer is 314 mm2.   D) 1256 mm2 Incorrect. It looks like you squared 20 and then multiplied by . 20 is the diameter, not the radius! To find the area, use the formula . The correct answer is 314 mm2.

 

 

 

Now that you know how to calculate the circumference and area of a circle, you can use this knowledge to find the perimeter and area of composite figures. The trick to figuring out these types of problems is to identify shapes (and parts of shapes) within the composite figure, calculate their individual dimensions, and then add them together.

 

For example, look at the image below. Is it possible to find the perimeter?

 

The first step is to identify simpler figures within this composite figure. You can break it down into a rectangle and a semicircle, as shown below.

 

You know how to find the perimeter of a rectangle, and you know how to find the circumference of a circle. Here, the perimeter of the three solid sides of the rectangle is 8 + 20 + 20 = 48 feet. (Note that only three sides of the rectangle will add into the perimeter of the composite figure because the other side is not at an edge; it is covered by the semicircle!)

 

To find the circumference of the semicircle, use the formula  with a diameter of 8 feet, then take half of the result. The circumference of the semicircle is , or approximately 12.56 feet, so the total perimeter is about 60.56 feet.

 

 

 

 

 

 

 

What is the area (to the nearest hundredth) of the figure shown below? (Both rounded regions are semi-circles.)     A) 16.56 in2 B) 7.14 in2 C) 4 in2 D) 3.14 in2   Show/Hide Answer A) 16.56 in2 Incorrect. It looks like you calculated the area of a circle using a radius of 2; in this figure, the radius of each circle is 1. To find the area of the figure, imagine the two semi-circles are put together to create one circle. Then calculate the area of the circle and add it to the area of the square. The correct answer is 7.14 in2.   B) 7.14 in2 Correct. Imagine the two semi-circles being put together to create one circle. The radius of the circle is 1 inch; this means the area of the circle is . The area of the square is 2 • 2 = 4. Adding those together yields 7.14 in2.   C) 4 in2 Incorrect. It looks like you calculated the area of the square, but not the circle. Imagine the two semi-circles are put together to create one circle. Then calculate the area of the circle and add it to the area of the square. The correct answer is 7.14 in2.   D) 3.14 in2 Incorrect. It looks like you calculated the area of the circle, but not the square. Calculate the area of the square and add it to the area of the circle. The correct answer is 7.14 in2.

 

 

 

Circles are an important geometric shape. The distance around a circle is called the circumference, and the interior space of a circle is called the area. Calculating the circumference and area of a circle requires a number called pi (), which is a non-terminating, non-repeating decimal. Pi is often approximated by the values 3.14 and . You can find the perimeter or area of composite shapes—including shapes that contain circular sections—by applying the circumference and area formulas where appropriate.