LESSON What is the relationship between the circumference of a circle and its diameter? Ratios and Pi 4.4
Texas Essential Knowledge and Skills The student is expected to: Proportionality—7.5.B Describe as the ratio of the circumference of a circle to its diameter. Mathematical Processes 7.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Warm Up The length and width of a rectangle are each multiplied by 5. Find how the perimeter and area of the rectangle change. The perimeter is multiplied by 5, and the area is multiplied by 25.
A circle is the set of all points in a plane that are the same distance from a given point, called the center. Center
A line segment with one endpoint at the center of the circle and the other endpoint on the circle is a radius (plural: radii). Center Radius
A diameter is a line segment that passes through the center of the circle and has both endpoints on the circle. The length of the diameter is twice the length of the radius. Center Radius Diameter
Additional Example 1: Naming Parts of a Circle Name the circle, a diameter, and three radii. N The circle is circle Z.LM is a diameter.ZL, ZM, and ZN are radii. M Z L
Check It Out: Example 1 Name the circle, a diameter, and three radii. The circle is circle D. IG is a diameter.DI, DG, and DH are radii. G H D I
The distance around a circle is called the circumference. Center Radius Diameter Circumference
The ratio of the circumference to the diameter,, is the same for any circle. This ratio is represented by the Greek letter , which is read “pi.” C d C d =
The formula for the circumference of a circle is C = d, or C = 2r. The decimal representation of pi starts with and goes on forever without repeating. We estimate pi using either 3.14 or. 22 7
Additional Example 2: Application A skydiver is laying out a circular target for his next jump. Estimate the circumference of the target by rounding to 3. C = dC 3 8C 24 ft Write the formula. Replace with 3 and d with 8. 8 ft
Check It Out: Example 2 A second skydiver is laying out a circular target for his next jump. Estimate the circumference of the target by rounding to 3. C = dC 3 14C 42 yd Write the formula. Replace with 3 and d with yd
Additional Example 3A: Using the Formula for the Circumference of a Circle Find the missing value to the nearest hundredth. Use 3.14 for pi. d = 11 ft; C = ? C = dC C ft Write the formula. Replace with 3.14 and d with ft
Additional Example 3B: Using the Formula for the Circumference of a Circle Find each missing value to the nearest hundredth. Use 3.14 for pi. r = 5 cm; C = ? C = 2rC C 31.4 cm Write the formula. Replace with 3.14 and r with 5. 5 cm
Additional Example 3C: Using the Formula for the Circumference of a Circle Find each missing value to the nearest hundredth. Use 3.14 for pi. C = cm; d = ? C = d 3.14d7.00 cm d Write the formula. Replace C with and with d _______ Divide both sides by 3.14.
Check It Out: Example 3A Find the missing value to the nearest hundredth. Use 3.14 for pi. d = 9 ft; C = ? C = dC C ft Write the formula. Replace with 3.14 and d with 9. 9 ft
Check It Out: Example 3B Find each missing value to the nearest hundredth. Use 3.14 for pi. r = 6 cm; C = ? C = 2rC C cm Write the formula. Replace with 3.14 and r with 6. 6 cm
Check It Out: Example 3C Find each missing value to the nearest hundredth. Use 3.14 for pi. C = cm; d = ? C = d 3.14d6.00 cm d Write the formula. Replace C with and with d _______ Divide both sides by 3.14.
ADDITIONAL EXAMPLE 1 Determine if the radius and the diameter of the two circles are proportional. The two circles are proportional. A.
ADDITIONAL EXAMPLE 1 Determine if the radius and the diameter of the two circles are proportional. The two circles are proportional. B.
ADDITIONAL EXAMPLE 2 The circumference of the smaller circle is 62.8 in. Use a proportion to find the circumference of the larger circle. 157 in A.
ADDITIONAL EXAMPLE 2 A smaller circle has a circumference of m and a diameter of 7 m. The circumference of a larger circle is m. Use a proportion to find the diameter of the larger circle. 21 m B.
4.4 LESSON QUIZ An artist is making a mobile using two sizes of circles. The smaller size has a 3 in. diameter and a 9.42 in. circumference. The larger size has a 5 in. diameter. What is the circumference of the larger size? 7.5.B in yards A circular well has a circumference of 7.85 yards. What is the diameter of the well? (Remember = 3.14.)
The circumference of the larger circle is ft, and the circumference of the smaller circle is ft. Use a proportion to find the diameter of the larger circle ft
The purple and blue design shown uses parts of a large circle and 2 identical smaller circles. If the diameter of the large circle is 18 inches, find the perimeter of the purple part of the design. Explain.
56.52 inches; The perimeter of the purple part of the design is made of half the circumference of one smaller circle, half the circumference of the other smaller circle, and half the circumference of the larger circle. Since the two smaller circles are identical, the perimeter is equal to the sum of the circumference of a smaller circle, and half the circumference of the larger circle. Since the ratio of the circumference of a circle to its diameter is equal to, we can write and solve the equation = 3.14.
The circumference of the large circle is inches, and half the circumference is inches. The diameter of the smaller circles is half that of the larger circle, so to find the circumference of the smaller circle we can write and solve the equation = The circumference of the smaller circle is inches. Add to find the perimeter of the purple section: = inches.
What is the relationship between the circumference of a circle and its diameter? Sample answer: The ratio of a circle’s circumference to its diameter is a constant, pi. Pi can be estimated as 3.14 or.