Circumference
In this activity, you will learn to find the circumference of a circle and explore the ratio of the circumference to the diameter. This ratio is called π.
1. In the Gizmotm, select the CIRCLE tab. Then use the slider to set Radius (r) to 4.0. (To quickly set a value, type a number in the box to the right of the slider and press ENTER.)
a. Which line segment in this circle is 4.0 units long? ______(Note: This segment is not currently displayed in the Gizmo.)
b. What is the length of the diameter in this circle?______Click on Show diameter to check your answer.
c. Write a formula to find d, the diameter, given the radius, r. ______
d. What is a formula to find r, given d?______
2. Click on Click to measure arc lengths to turn on the arc length ruler. An arc length is the measure of part of a circle.
a. Use an arc length ruler to measure the full circumference, which is the distance around the circle. Attach both handles of the arc length ruler to point B. What is the circumference of this circle? ______
b. Use a calculator to divide the circumference (C) by the diameter (d). What is your answer?______Click on Show circumference info to check your work. The answer should be about 3.14159, which is an approximate value for π (the Greek letter pi).
c. With the arc length ruler still attached and Show circumference turned on, drag B to make circles of different sizes. Notice the values for r, C, and . What changes? ______What stays the same? ______In a circle, what is always 2r units long? ______
3. Turn off Show circumference info and the arc length ruler. Instead of measuring the distance around the circle, you can quickly find the circumference of a circle by using the formula C = 2πr. Since you know that 2r = d, you can also use C = πd.
a. Use a formula to find C when r = 5.0. Which formula will you choose? ______Use the π button on your calculator to find the circumference. Round your answer to the nearest tenth. ______Then turn on Show circumference info to check your answer.
b. On your own, find the circumference of a circle with d = 25. Which formula is best to use in this case? Formula used:______C=______Check your answer in the Gizmo.
c. When r is a multiple of 7, or is a fraction, it is sometimes easier to use as an estimate for π. Use this value for π to find C when r = 14. What is your answer? ______
Area of a Circle
In this activity, you will learn to find the area of a circle and explore the ratio of the area to the square of the radius.
1. With Show diameter, Show circumference info, and the arc length ruler all turned off, set Radius (r) = 6.0. Click on Show area info.
2. The formula for the area of a circle is A = πr2. Which value do you use in this formula, the radius or the diameter?______What does r2 mean? ______
3. Use a calculator and the formula A = πr2 to find the area of this circle. (Be careful. Only r gets squared, not π.) ______Round your answer to the nearest tenth. Check your answer using the Gizmo. Correct? Yes or No. ______
4. Find the area, rounded to the nearest tenth, of a circle with diameter of 15 meters. What do you need to do first, in order to use the formula A = πr2? ______Click on Show diameter to be sure that you are using the correct value for r. Use the Gizmo to check your answer for the area of the circle.
5. With Show area info turned on, turn off Show diameter.
6. Under Show area info, notice the column. What is the value of ?______What is another name for this value? ______
7. Drag point B to make different size circles. Notice the values for r, A, and . What changes?______What stays the same? ______