Lesson 16: The Most Famous Ratio of All

Lesson 16: The Most Famous Ratio of All

Discovering pi.

Opening Exercise

• 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.

• Write your own definition for the term circle. 

• Extend segment 𝐶𝐵 to a segment 𝐴𝐵 in part (a), where 𝐴 is also a point on the circle.

Multiple Choice

The length of the segment 𝐴𝐵 is called the diameter of the circle. The diameter is ______________________ as long as the radius

• half

• twice

• triple

• a little more than 3

Measure the radius and diameter of each circle. The center of each is labeled 𝐶.

Multiple Choice

Which of these have the radii correctly listed from least to greatest?

• 1cm, 2cm, 3cm

• 1.5cm, 2.5cm, 3cm

• 1.5cm, 2cm, 3cm

• 2cm, 2.5cm, 3cm

Multiple Choice

The radius of the largest circle is 𝟑 centimeters. What is the diameter?

• 3 cm

• 4 cm

• 5 cm

• 6 cm

• 9 cm

Multiple Select

If we draw a circle with a radius of 6 centimeters using a compass what is the measurement of the diameter and why?

• 3cm

• 6cm

• 12cm

• because it is half

• because it is twice

Pi Activity

Finding

 $\pi$  

Mathematical Modeling Exercise 

• The ratio of the circumference to its diameter is always the same for any circle.

• The value of this ratio, Circumference/Diameter, 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

Pi is a non-terminating, non-repeating decimal approximations for pi are-

• 3.14

•  $\frac{22}{7}$  

• 3.14159265359......

• Circumference of a Circle = 𝜋 × Diameter.

Find the circumference of each circle

•  $Use\ \frac{22}{7}\ as\ an\ approximation\ of\pi.$  

• 𝐂𝐢𝐫𝐜𝐮𝐦𝐟𝐞𝐫𝐞𝐧𝐜𝐞 𝐨𝐟 𝐚 𝐂𝐢𝐫𝐜𝐥𝐞 = 𝝅 × 𝐃𝐢𝐚𝐦𝐞𝐭𝐞r.

Multiple Select

Choose the 3 correct answers listed below.

• 33cm

• 66cm

• 268ft

• 286ft

• 110m

Multiple Choice

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.)

• 36.7cm

• 36.8cm

• 73.4cm

• 73.5cm

Multiple Choice

The radius of a paper plate is 11.7 cm. Find the circumference to the nearest hundredth. Use the pi button on a calculator.

• 36.74cm

• 73.48cm

• 73.51cm

• 73.52cm

Multiple Select

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. Choose 2.

•  $C=\pi2r$  
  

•  $C=2\pi2r$  

•  $C=2\pi r$  

•  $C=\pi d$  

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.)

Show your work in your module

Multiple Choice

What is the perimeter of the semicircle?

• 20.56m

• 25.12m

• 33.12m

• 12.56m

Relevant Vocabulary 

• CIRCLE: Given a point 𝑂 in the plane and a number 𝑟 > 0, the circle with center 𝑂 and radius 𝑟 is the set of all points in the plane whose distance from the point 𝑂 is equal to 𝑟.

• 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.

Relevant Vocabulary 

• 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. 

• 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

Exit Ticket

The Exit Ticket calls on students to synthesize their knowledge of circles and rectangles. 

Multiple Select

Brianna’s parents built a swimming pool in the backyard. Brianna says that the distance around the pool is 120 feet. Why or why not?

• yes

• no

• she included the semicircle

• she only measured the rectangle

Multiple Select

How could Brianna determine the distance around the pool so that her parents would know how many feet of stone to buy for the edging around the pool.

• First find half the circle and then add 120.

• First find half the circle and then add 100.

• C=1/2(pi) x 20 + 120

• C= 1/2(pi x 20) + 100

Multiple Choice

What is the relationship between the circumference of the semicircular part of the pool and the width of the pool?

• twice the diameter

• it is half of pi because it is half the circle

• it is 1.5 times the diameter

• no relationship

This is the end my friend

Do the Problem Set on page 100