Circumference of a Circle
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1 Circumference of a Circle A circle is a shape with all points the same distance from the center. It is named by the center. The circle to the left is called circle A since the center is at point A. If you measure the distance around a circle and divide it by the distance across the circle through the center, you will always come close to a particular value, depending upon the accuracy of your measurement. This value is approximately We use the Greek letter (pronounced Pi) to represent this value. The number goes on forever. However, using computers, mathematicians have been able to calculate the value of to thousands of places. The distance around a circle is called the circumference. The distance across a circle through the center is called the diameter. is the ratio of the circumference of a circle to the diameter. Thus, for any circle, if you divide the circumference by the diameter, you get a value close to. This relationship is expressed in the following formula: where is circumference and is diameter. You can test this formula at home with a round dinner plate. If you measure the circumference and the diameter of the plate and then divide by, your quotient should come close to. Another way to write this formula is: where means multiply. This second formula is commonly used in problems where the diameter is given and the circumference is not known (see the examples below). The radius of a circle is the distance from the center of a circle to any point on the circle. If you place two radii end-to-end in a circle, you would have the same length as one diameter. Thus, the diameter of a circle is twice as long as the radius. This relationship is expressed in the following formula:, where is the diameter and is the radius. Circumference, diameter and radii are measured in linear units, such as inches and centimeters. A circle has many different radii and many different diameters, each passing through the center. A real-life example of a radius is the spoke of a bicycle wheel. A 9-inch pizza is an example of a diameter: when one makes the first cut to slice a round pizza pie in half, this cut is the diameter of the pizza. So a 9- inch pizza has a 9-inch diameter. Let's look at some examples of finding the circumference of a circle. In these examples, we will use = 3.14 to simplify our calculations. Example 1: The radius of a circle is 2 inches. What is the diameter? = 2 (2 in) = 4 in Example 2: The diameter of a circle is 3 centimeters. What is the circumference?
2 = 3.14 (3 cm) = 9.42 cm Example 3: The radius of a circle is 2 inches. What is the circumference? = 2 (2 in) = 4 in = 3.14 (4 in) = in Example 4: The circumference of a circle is 15.7 centimeters. What is the diameter? 15.7 cm = cm 3.14 = = 15.7 cm 3.14 = 5 cm Summary: The number is the ratio of the circumference of a circle to the diameter. The value of is approximately The diameter of a circle is twice the radius. Given the diameter or radius of a circle, we can find the circumference. We can also find the diameter (and radius) of a circle given the circumference. The formulas for diameter and circumference of a circle are listed below. We round to 3.14 in order to simplify our calculations.
3 Area of a Circle Step 1: Using the compass, draw a circle of radius 7 cm. Then mark the circle's centre and draw its radius. Step 2: Place the centre of the protractor at the centre of the circle and the zero line along the radius. Then mark every 30º around the circle. Step 3: Using a ruler and a pencil, draw lines joining each 30º mark to the centre of the circle to form 6 diameters. The diagram thus obtained will have 12 parts as shown below.
4 Step 4: Colour the parts as shown below. Step 5: Cut out the circle and then cut along the diameters so that all parts (i.e. sectors) are separated. Step 6: Arrange all of the sectors to make a shape that approximates a parallelogram as shown below. Step 7: Using a ruler, measure the base and the height of the approximate parallelogram obtained in Step 6. Questions: 1. Calculate the area of the figure in Step 6 by using the formula: 2. What is the area of the circle drawn in Step 1? 3. It appears that there is a formula for calculating the area of a circle. Can you discover it? Formula for the Area of a Circle
5 From the above activity, it is clear that by arranging the sectors of the circle as a parallelogram that: Remember: The area, A, of a circle is given by the following formula where r is the radius of the circle:
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