Alg/Geo



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 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.160020076200200977516510These are formulas that you have used in previous courses. Let’s take a look at their derivations:I. Circumference of a circle:Circumference ?E 20.420 cm32956502540Circumference ?A = 13.509 cm333375-3810 FM = 6.5 cm BC = 4.3cmCalculate the ratio of the circumference to the diameter for each of the above circles.Notes: OVER Page 2A.More information about π:π is an irrational number which meansExamples of other irrational numbers: Approximations for π: 3,3 1/8, 3.14, 3.14159, 22/7B.Exact and approximate answers:Circle the expression which represents the exact circumference of a circle with a diameter of 8 units.8 * 3.148 * 3.141598 π 8 * 22/7C.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; diameterII.Area of a circle:Method 1: Successively Smaller Sectors48577516510 Page 3Method 2: Unrolling Concentric Circles: Method 3: Regular Polygons with Infinitely Many Sides:380047511620531432520955A.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.OVER Page 4Lesson 6-4 HomeworkShow 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.742950711204. 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.Page 57.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 ft9.The semicircular window below has a radius of 4 ft.48577523495a.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. 933450137795 OVER Page 6 14. If the circumference of a circle measures 10π feet, what is the area in square feet?838200154305 15.What is the area, in ft2, of the shaded region if the area of Circle E is 3ft2 and ABCD is a square?21526497556483820013271516.A circular table top has a radius of 2/3 yd. What is the area of the table top? 761999115570 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. ................
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