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Circumference of Circles and Arc Length

Circumference of Circles and Arc Length. Concept 56. CCSS. Content Standards

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Circumference of Circles and Arc Length

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  1. Circumference of Circles and Arc Length Concept 56

  2. CCSS Content Standards G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Mathematical Practices 1 Make sense of problems and persevere in solving them. 6 Attend to precision

  3. Circumference – the distance around a circle (perimeter of a circle).

  4. 1. A series of crop circles was discovered in Alberta, Canada, on September 4, 1999. The largest of the three circles had a radius of 30 feet. Find its circumference. Since the radius is 30 feet, and d = 2r, the diameter = 2(30) or 60 feet. C = dCircumference formula = (60) Substitution = 60 Simplify. ≈ 188.50 Use a calculator. Answer: C= 60 feet or about 188.50 feet.

  5. 2. The Unisphere is a giant steel globe that sits in Flushing Meadows-Corona Park in Queens, New York. It has a diameter of 120 feet. Find its circumference. A. 377.0 feet B. 392.5 feet C. 408.3 feet D. 422.1 feet

  6. Divide each side by . 3. Find the diameter and the radius of a circleto the nearest hundredth if the circumference of the circle is 65.4 feet. Circumference Formula Substitution Use a calculator. Radius Formula Use a calculator. Answer:d ≈ 20.82 ft; r ≈ 10.41 ft

  7. 4. Find the radius of a circle to the nearest hundredth if its circumference is 16.8 meters. A. 8.4 m B. 5.35 m C. 2.67 m D. 16.8 m

  8. 5. You need to find the diameter of the circle and use it to calculate the circumference. The radius of the circle is the same length as either leg of the triangle. The legs of the triangle have equal length. Call the length x. So the radius of the circle is 3.

  9. A. B. C. D. 6.

  10. Arc length – the length in linear measurement of an arc. A portion of the circle’s circumference. The ratio of the length of an arc ( ) to the circumference ( ) of the circle is equal to the ratio of the degree measure ( ) of the arc to 360. c l x∘

  11. cm Use a calculator. Answer: 7. Arc Length Equation Substitution

  12. Answer: 8. Arc Length Equation Substitution Use a calculator.

  13. Answer: 9. Arc Length Equation Substitution Use a calculator.

  14. 10. A. 3.56 cm B. 3.77 cm C. 3.98 cm D. 4.21 cm

  15. 11. A. 36.56 cm B. 37.79 cm C. 38.61 cm D. 40.21 cm

  16. 12. A. 32.99 cm B. 33.59 cm C. 33.89 cm D. 34.61 cm

  17. Area of Circles and Sectors Concept 57

  18. 13. Find the area of the circle.

  19. 14. Find the area of the circle.

  20. 15. MANUFACTURING An outdoor accessories company manufactures circular covers for outdoor umbrellas. If the cover is 8 inches longer than the umbrella on each side, find the area of the cover in square inches. The diameter of the umbrella is 72 inches, and the cover must extend 8 inches in each direction. So, the diameter of the cover is 8 + 72 + 8 or 88 inches. Divide by 2 to find that the radius is 44 inches.

  21. 16. A swimming pool company manufactures circular covers for above ground pools. If the cover is 1 foot longer than the pool on each side, find the area of the cover. A. 62.8 ft2 B. 254.5 ft2 C. 314.2 ft2 D. 1256.6 ft2

  22. Take the positive square root of each side. = r 15. Find the radius of a circle with an area of 58 square inches. Area of a circle Substitution Divide each side by . 4.3 ≈ r Simplify. Answer: The radius of the circle is about 4.3 in.

  23. 16. Find the radius of a circle with an area of 45 square inches. A. 3.8 in. B. 4.5 in. C. 5.7 in. D. 7.6 in.

  24. 19. Find the circumference of each circle. Use your calculator's value of [pi]. Round your answer to the nearest tenth.

  25. sector of a circle – a section of the circle made or bounded by a central angle and its intercepted arc

  26. Concept 2

  27. Find the area of each sector. 21. 20.

  28. 22. A pie has a diameter of 9 inches and is cut into 10 congruent slices. What is the area of one slice to the nearest hundredth? 9 in Answer: The area of one slice of pie is about6.36 square inches.

  29. 23.A pizza has a diameter of 14 inches and is cut into 8 congruent slices. What is the area of one slice to the nearest hundredth? A. 16.21 in2 B. 19.24 in2 C. 26.43 in2 D. 38.48 in2

  30. 24. Find the area of the shaded section.

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