Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Your teacher will give your group two circles of different sizes.
For each circle,
Use the grid squares to determine the diameter of the circle.
Estimate the area of the circle.
Record your measurements in the first two rows of the table.
diameter (cm)
estimated area (cm2)
Plot your diameter and area values on the coordinate plane. What do you notice?
A coordinate plane with the origin labeled O. The horizontal axis is labeled "diameter, in centimeters," and the numbers 5 through 25, in increments of 5, are indicated. The vertical axis is labeled "area, in square centimeters, and the numbers 50 through 300, in increments of 50, are indicated.
Find out the measurements from another group that measured different circles. Record their values in your table, and plot them on your same coordinate plane.
Earlier, you graphed the relationship between the diameter and circumference of a circle. How is this graph the same? How is it different?
7.3
Activity
Covering a Circle
Standards Alignment
Building On
Addressing
7.G.A
Draw, construct, and describe geometrical figures and describe the relationships between them.
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Here is a square whose side length is the same as the radius of the circle.
How many of these squares do you think it would take to cover the circle exactly?
Student Lesson Summary
The circumference of a circle is proportional to the diameter , and we can write this relationship as . The circumference is also proportional to the radius of the circle, and the constant of proportionality is because the diameter is twice as long as the radius. However, the area of a circle is not proportional to the diameter (or the radius).
The area of a circle with radius is a little more than 3 times the area of a square with side so the area of a circle of radius is approximately . We saw earlier that the circumference of a circle of radius is . If we write for the circumference of a circle, this proportional relationship can be written .
The area of a circle with radius is approximately . Unlike the circumference, the area is not proportional to the radius because cannot be written in the form for a number . We will investigate and refine the relationship between the area and the radius of a circle in future lessons.
Glossary
area of a circle
The area of a circle is the number of square units it covers. If a circle has radius units, its area is square units.
For example, a circle has radius of 3 inches. Its area is , or , square inches. This is about 28.3 square inches.
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