Math 2 Name ______
Lesson 6-4: Circumference and Area of Circles
Learning Goals:
· I can use algebra to demonstrate that because π is the ratio of a circle’s circumference to its diameter that the formula for a circle’s circumference must be C=π∙d.
· I can use pictures to explain that a regular polygon with many sides is nearly a circle, its perimeter is nearly the circumference of a circle, and that its apothem is nearly the radius of a circle.
· I can substitute the “nearly” values of a many sided regular polygon into
A=12∙apothem∙perimeter to show that the formula for the area of a circle is A=π∙r2.
· I can set up and solve equations involving the formulas for circumference and area of a circle.
Before you start this activity, from the home screen on your calculator:
5: Settings à 2: Document Settings
Set the “Display Digits” to “Float”
Select “Make Default”
These are formulas that you have used in previous courses. Let’s take a look at their derivations:
I. Circumference of a circle:
Circumference ʘE » 20.420 cm
Circumference ʘA = 13.509 cm
BC = 4.3cm FM = 6.5 cm
Ø Calculate the ratio of the circumference to the diameter for each of the above circles.
Notes: https://learnzillion.com/lesson_plans/6728-relate-diameter-and-circumference
A. More information about π:
π is an irrational number which means
Examples of other irrational numbers:
Some approximations for π are: 3, 3 1/8, 3.14, 3.14159, 22/7
B. Exact versus approximate answers:
Circle the expression which represents the exact circumference of a circle with a diameter of 8 units.
8 * 3.14 8 * 3.14159 8 π 8 * 22/7
C. Examples:
1. Find the circumference of a circle with radius 23 cm.
Use the button when calculating approximate answers!
Exact ______To the nearest tenth of a cm ______
2. The circumference of a circle is given. Find the desired length.
a. 242 cm; diameter b. 8.75 ft; radius c. 36π in; diameter
II. Area of a circle: https://learnzillion.com/lesson_plans/6870-informally-prove-the-area-of-a-circle
Method 1: Successively Smaller Sectors
Notes:
Method 2: Unrolling Concentric Circles Method 3: Regular Polygons with
Infinitely Many Sides:
A. Examples:
1. Find the area of a circle with radius 18 cm.
Exact: ______Round to the nearest tenth: ______
2. Find the area of a circle with diameter of 18 cm.
Exact: ______Round to the nearest hundredth: ______
3. The area of a circle is 225π ft2. 4. The area of a circle is 50 in2.
Find the exact value of the radius. Find its diameter to the nearest 100th of an inch.
5. Find the circumference of the circle if the square has a perimeter of 36 cm.
Math 2 Name ______
Lesson 6-4: Circumference and Area of Circles Practice
Show your work for the following problems. Label your answers with the appropriate units. Give both the exact and approximate answer. Round to the nearest 100th unless otherwise indicated.
1. A 16-foot rope is attached to a metal stake that is driven into the ground. A large dog is tied to the other end of the rope.
a. The dog walks around in circles with the leash fully extended. How far does the dog walk in one full rotation around the stake?
b. What is the area that the dog can roam?
c. If the rope were increased by one foot, how much longer would the dog walk in one lap around and how much more area would the dog have to roam?
2. The diagonal of the below square is 40 inches. Find the circumference of the circle.
3. The radius of a circle is 14 in. What is the circumference of the circle?
A. 87.96 in. B. 615.44 in. C. 43.96in. D. 21.98 in.
4. The circumference of a circle is 22 cm. What is the diameter?
A. 7 cm. B. 69.08 cm. C. 11 cm. D. 3.5 cm.
5. Which expression shows the exact area of a circle with a radius of 18 miles?
A. 324 * 3.14 mi2 B. 36π mi2 C. 324π mi2 D. 1017.87601976 mi2
6. The area of a circle is 625π ft2. What is its radius?
A. 312.5 ft B. 25 ft C. 198.94 ft D. 14.10 ft
7. The semicircular window below has a radius of 4 ft.
a. Find the length of the window.
b. Find the area of the window.
8. A radio tower can transmit a call anywhere within a 50-mile radius from the transmission point. What is the area of the transmission region?
9. A sprinkler spins 360° and has a 5.6-foot spray. Find the area of the lawn the sprinkler will water.
10. The area of the square is 324 ft2. Find the area of the shaded region.
11. If the circumference of a circle measures 10π feet, what is the area in square feet?
12. A circular table top has a radius of 2/3 yd. What is the area of the table top?
13. Which has the greater area, a circle with a diameter of 4 feet or a square with a side of 4 feet? Justify your answer.
EXTENSION:
What is the area, in ft2, of the shaded region if the area of Circle E is 3ft2 and ABCD is a square?