The Most Famous Ratio of All - Pi

Video solutions to help Grade 7 students learn how to how to develop the definition of circle using diameter and radius.

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New York State Common Core Math Grade 7, Module 3, Lesson 16

Lesson 16 Student Outcomes

• Students learn how to develop the definition of circle using diameter and radius.
• Students know that the distance around a circle is called the circumference and discover that the ratio of the circumference to the diameter of a circle is a special number called pi, written π.
• Students know the formula for the circumference C of a circle of diameter d and radius r. They use scale models to derive these formulas.
• Students use 22/7 and 3.14 as estimates for π and informally show that is slightly greater than 3.

Relevant Vocabulary

Circle: Given a point C in the plane and a number r > 0, the circle with center C and radius r is the set of all points in the plane that are distance r from the point C.

Radius of a circle: The radius is the length of any segment whose endpoints are the center of a circle and a point that lies on the circle.

Diameter of a circle: The diameter of a circle is the length of any segment that passes through the center of a circle whose endpoints lie on the circle. If r is the radius of a circle, then the diameter is 2r. The word diameter can also mean the segment itself. Context determines how the term is being used: "the diameter" usually refers to the length of the segment, while "a diameter" usually refers to a segment. Similarly, "a radius" can refer to a segment from the center of a circle to a point on the circle.

Circumference: The circumference of a circle is the distance around a circle.

Pi: The number pi, denoted by π, is the value of the ratio given by the circumference to the diameter. The most commonly used approximations for π is 3.14 or 22/7.

Semicircle: Let C be a circle with center O, and let A and B be the endpoints of a diameter. A semicircle is the set containing A, B, and all points that lie in a given half-plane determined by AB (diameter) that lie on circle C.

Lesson 16 Problem Set Classwork

Opening Exercise

a. Using a compass, draw a circle like the picture to the right.
C is the center of the circle.
The distance between C and B is the radius of the circle.
b. Write your own definition for the term circle.
c. Extend segment CB to a segment AB, where A is also a point on the circle.
The length of the segment AB is called the diameter of the circle.
d. The diameter is twice, or 2 times, as long as radius.
e. Measure the radius and diameter of each circle. The center of each circle is labeled C.
f. Draw a circle of radius 6 cm.

Example 1
The ratio of the circumference to its diameter is always the same for any circle. The value of this ratio, is called the number pi and is represented by the symbol π.
Since the circumference is a little greater than 3 times the diameter, π is a number that is a little greater than 3. State:
Use the π symbol to represent this special number. Pi is a non-terminating, non-repeating decimal and mathematicians use the symbol π or approximate representations as more convenient ways to represent pi.

Example 2
a. The following circles are not drawn to scale. Find the circumference of each circle. (Use 22/7 as an approximation for π.)
b. The radius of a paper plate is 11.7 cm. Find the circumference to the nearest tenth. (Use 3.14 as an approximation for π.)
c. The radius of a paper plate is 11.7 cm. Find the circumference to the nearest hundredth. ((Use the button on your calculator as an approximation for π.)
d. A circle has a radius of r cm and a circumference of C cm. Write a formula that expresses the value of C in terms of r and π.
e. The figure below is in the shape of a semicircle. A semicircle is an arc that is "half" of a circle. Find the perimeter of the shape. (Use 3.14 for π.)

Lesson 16 Problem Set Sample Solutions

1. Find the circumference.
a. Give an exact answer in terms of π
b. Use π = 22/7 and express your answer as a fraction in lowest terms.

3. The figure shows a circle within a square. Find the circumference of the circle. Let π ≈ 3.14.

5. Find the perimeter of the semicircle. Let π ≈ 3.14.

7. Mary and Margaret are looking at a map of a running path in a local park. Which is the shorter path from E to F: along the two semicircles or along the larger semicircle? If one path is shorter, how much shorter is it?