Math 2Name ______
Lesson 6-4: Circumference and Area of CirclesDate ______
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 .
- 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
to show that the formula for the area of a circle is .
- I can set up and solve equations involving the formulas for circumference and area of a circle.
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
FM = 6.5 cm
BC = 4.3cm
Calculate the ratio of the circumference to the diameter for each of the above circles.
Notes:
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A.More information about π:
π is an irrational number which means
Examples of other irrational numbers:
Approximations for π: 3,3 1/8, 3.14, 3.14159, 22/7
B.Exact and approximate answers:
Circle the expression which represents the exact circumference of a circle with a diameter of 8 units.
8 * 3.148 * 3.141598π 8 * 22/7
C.Examples:
1.Give the circumference of a circle with radius 23 cm.
Exact ______To the nearest 10th of a cm ______
2.The circumference of a circle is given. Find the desired length.
a.242 cm; diameterb. 8.75 ft; radiusc.36π in; diameter
II.Area of a circle:
Method 1: Successively Smaller Sectors
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Method 2: Unrolling Concentric Circles: Method 3: Regular Polygons with
Infinitely Many Sides:
A.Examples:
1.Give the area of a circle with radius 18 cm.
Exact: ______Round to the nearest 10th: ______
2.Give the area of a circle with diameter of 18 cm.
Exact: ______Round to the nearest 10th: ______
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.
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Lesson 6-4 Homework
Show your work for the following problems. Label your answers with the appropriate units & round to the nearest 100th, unless otherwise indicated.
- 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.
- The dog walks around in circles with the leash fully extended. How far does the dog walk in one full rotation around the stake?
- What is the area that the dog can roam?
- If the rope were increased by one foot, what would be the new distance around and the new area?
- Find the circumference of the circle if the square has a perimeter of 36 cm.
3.The diagonal of the below square is 40 inches. Find the circumference of the circle.
4. The diameter of a circle is 7 ft. What is the circumference of the circle?
A. 3.5 ft.B. 14 ft.C. 21.99 ft.D. 38.47 ft.
5. 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.
6. 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.
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7.Which expression shows the exact area of a circle with a radius of 18 miles?
A.324 * 3.14 mi2B.36π mi2C.324π mi2D.1017.87601976 mi2
8.The area of a circle is 625π ft2. What is its radius?
A.312.5 ftB.25 ftC.198.94 ftD.14.10 ft
9.The semicircular window belowhas a radius of4 ft.
a.Find the length of the window.
b.Find the area of the window.
10.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?
11.A circle has an area of 256.38 cm2. What is its radius?
12.A sprinkler spins 360° and has a 5.6-foot spray. Find the area of the lawn the sprinkler will water.
13.The area of the square is 324 ft2. Find the area of the shaded region.
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14. If the circumference of a circle measures 10π feet, what is the area in square feet?
15.What is the area, in ft2, of the shaded region if the area of Circle E is 3ft2 and ABCD is a square?
16.A circular table top has a radius of 2/3 yd. What is the area of the table top?
17.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.