11.4 Circumference and Arc Length
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1 11.4 ircumference and rc Length Goal p Find arc lengths and other measures. Your Notes VOULRY ircumference rc length THEOREM 11.8: IRUMFERENE OF IRLE The circumference of a circle is r 5 or 5, where d d is the diameter of the circle and r is the radius of the circle. 5 5 Example 1 Use the formula for circumference Find the indicated measure. a. ircumference of a circle b. Radius of a circle with with radius 11 meters circumference 18 yards a. 5 2πr b. 5 2πr 5 2 p π p 5 2πr 5 π 5 r < m yd < r heckpoint omplete the following exercise. 1. Find the circumference of a circle with diameter 23 inches. 306 Lesson 11.4 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.
2 11.4 ircumference and rc Length Goal p Find arc lengths and other measures. Your Notes VOULRY ircumference The circumference of a circle is the distance around the circle. rc length n arc length is a portion of the circumference of a circle. THEOREM 11.8: IRUMFERENE OF IRLE The circumference of a circle is r 5 πd or 5 2πr, where d d is the diameter of the circle and r is the radius of the circle. 5 πd 5 2πr Example 1 Find the indicated measure. a. ircumference of a circle b. Radius of a circle with with radius 11 meters circumference 18 yards a. 5 2πr b. 5 2πr 5 2 p π p πr 5 22 π Use the formula for circumference 18 } 2π 5 r < m 2.86 yd < r heckpoint omplete the following exercise. 1. Find the circumference of a circle with diameter 23 inches. about inches 306 Lesson 11.4 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.
3 lways pay attention to units. s in Example 2, you may need to convert units to get a correct answer. Example 2 Skateboarding The dimensions of the skateboard wheel shown at the right are in millimeters. To the nearest meter, how far does the wheel travel when it makes 35 revolutions? Step 1 Find the diameter of the wheel. d 5 1 2( ) 5 mm Step 2 Find the circumference of the wheel. 5 πd 5 π( ) < mm Step 3 Find the distance the wheel travels in 35 revolutions. In one revolution, the wheel travels a distance equal to its. In 35 revolutions, the wheel travels a distance equal to times its circumference. Distance traveled Use circumference to find distance traveled Number of 5 revolutions p ircumference < p mm 5 mm Step 4 Use unit analysis. hange meters. 1 m mm p } 1000 mm 5 m The wheel travels about meters millimeters to 20 heckpoint omplete the following exercise. 2. skateboard wheel has a diameter of 56 millimeters. How many revolutions does the wheel make when traveling 3 meters? opyright Holt McDougal. ll rights reserved. Lesson 11.4 Geometry Notetaking Guide 307
4 lways pay attention to units. s in Example 2, you may need to convert units to get a correct answer. Example 2 Skateboarding The dimensions of the skateboard wheel shown at the right are in millimeters. To the nearest meter, how far does the wheel travel when it makes 35 revolutions? Step 1 Find the diameter of the wheel. d ( 20 ) 5 55 mm Step 2 Find the circumference of the wheel. 5 πd 5 π( 55 ) < mm Step 3 Find the distance the wheel travels in 35 revolutions. In one revolution, the wheel travels a distance equal to its circumference. In 35 revolutions, the wheel travels a distance equal to 35 times its circumference. Distance traveled Use circumference to find distance traveled Number of 5 revolutions p ircumference < 35 p mm mm Step 4 Use unit analysis. hange 6048 millimeters to meters. 1 m 6048 mm p } 1000 mm m The wheel travels about 6 meters heckpoint omplete the following exercise. 2. skateboard wheel has a diameter of 56 millimeters. How many revolutions does the wheel make when traveling 3 meters? about 17 revolutions opyright Holt McDougal. ll rights reserved. Lesson 11.4 Geometry Notetaking Guide 307
5 R LENGTH OROLLRY In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to. rc length of }} 5 m } 2 π r, or r rc length of 5 m } p 2πr Example 3 Find the indicated measure. a. rc length of m Find and use arc lengths 2π1 2 b. m RS R 12.3 ft 38 ft a. rc length of 5 p 2π( ) < meters rc length of RS b. }} 5 m RS } Write equation. 2πr 5 m RS } Substitute. 2π1 2 p 5 m RS < m RS S Multiply each side by. Use a calculator. heckpoint Find the indicated measure. 3. rc length of 4. ircumference of (Z 4 ft X 978 Z cm Y 308 Lesson 11.4 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.
6 R LENGTH OROLLRY In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to. rc length of }} 5 m } 2 π r, or r rc length of 5 m } p 2πr Example 3 Find the indicated measure. a. rc length of m b. m RS R 12.3 ft 38 ft a. rc length of } p 2π( 2 ) < 3.07 meters rc length of RS b. }} 5 m RS } Write equation. 2πr 38 5 m RS } Substitute. 2π p Find and use arc lengths 38 2π m RS 1778 < m RS S Multiply each side by. Use a calculator. heckpoint Find the indicated measure. 3. rc length of 4. ircumference of (Z 4 ft 978 about 6.77 ft X Z cm Y cm 308 Lesson 11.4 Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.
7 Example 4 Use arc length to find distances Luggage conveyor belt for luggage at an airport is shown at the right. The outer part of the belt forms a 1808 arc at each end. For each arc, the radius is 8 feet. pproximate the distance around the belt for a coin on the outer portion. Round to the nearest foot. 8 ft 4 ft 20 ft The outer portion is made of two straight sections and two semicircles. To find the distance around the outer portion, find the sum of the lengths of each part. Distance 5 2 p Length of each straight section 5 2( ) 1 2 p p Length of each semicircle < feet The distance around the outer portion is about feet. heckpoint omplete the following exercise. 5. In Example 4, the inner portion of the belt also has 1808 arcs on each end. The radius of each arc is 4 feet. Find the distance around the belt for a coin on the inner portion. Round to the nearest foot. Homework opyright Holt McDougal. ll rights reserved. Lesson 11.4 Geometry Notetaking Guide 309
8 Example 4 Use arc length to find distances Luggage conveyor belt for luggage at an airport is shown at the right. The outer part of the belt forms a 1808 arc at each end. For each arc, the radius is 8 feet. pproximate the distance around the belt for a coin on the outer portion. Round to the nearest foot. 8 ft 4 ft 20 ft The outer portion is made of two straight sections and two semicircles. To find the distance around the outer portion, find the sum of the lengths of each part. Distance 5 2 p Length of each straight section 5 2( 20 ) 1 2 p 1 1 } 2 p 2π p p Length of each semicircle < feet The distance around the outer portion is about 90 feet. heckpoint omplete the following exercise. 5. In Example 4, the inner portion of the belt also has 1808 arcs on each end. The radius of each arc is 4 feet. Find the distance around the belt for a coin on the inner portion. Round to the nearest foot. about 65 ft Homework opyright Holt McDougal. ll rights reserved. Lesson 11.4 Geometry Notetaking Guide 309
9 Focus On Reasoning Use after Lesson 11.4 Your Notes Geometry on a Sphere Goal p ompare Euclidean and spherical geometries. VOULRY Great circle ROERTY SUMMRY OX Euclidean Geometry l lane contains line l and not on the line l. Spherical Geometry Sphere S contains enter and point not on m. m is a. m S m Example 1 ompare Euclidean and spherical geometry Is the following theorem true in spherical geometry? Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle is Draw a triangle on a globe with two points on the equator and the third at the North ole. The two sides of the triangle are perpendicular to the equator, ma 5 ma 5. Thus ma 1 ma 1 ma 1808 North ole Equator So, the Triangle Sum Theorem is not true in spherical geometry Focus on Reasoning Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.
10 Focus On Reasoning Use after Lesson 11.4 Your Notes Geometry on a Sphere Goal p ompare Euclidean and spherical geometries. VOULRY Great circle great circle is a circle on a sphere whose center is the center of the sphere. ROERTY SUMMRY OX Euclidean Geometry l lane contains line l and point not on the line l. Spherical Geometry Sphere S contains great circle m enter and point not on m. Great circle m is a line. S m Example 1 ompare Euclidean and spherical geometry Is the following theorem true in spherical geometry? Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle is Draw a triangle on a globe with two points on the equator and the third at the North ole. The two sides of the triangle are perpendicular to the equator, ma 5 ma Thus ma 1 ma 1 ma North ole Equator So, the Triangle Sum Theorem is not true in spherical geometry Focus on Reasoning Geometry Notetaking Guide opyright Holt McDougal. ll rights reserved.
11 Example 2 Find distances on a sphere The diameter of the sphere is 7 and m Find the distance between and. Find the lengths of the and the of the great circle. In each case let x be the arc length. rc length of }} 5 m x } } 5 } 2πr x < rc length of }} 5 m x } } 5 2πr 7 2 }} x < heckpoint omplete the following exercises. 1. What is the sum of the angles of an equilateral triangle on a sphere? 2. Find the distance between and. diameter 5 25, m Homework 3. Find the distance between and. diameter 5 30, m opyright Holt McDougal. ll rights reserved Focus on Reasoning Geometry Notetaking Guide 311
12 Example 2 Find distances on a sphere The diameter of the sphere is 7 and m Find the distance between and. Find the lengths of the minor arc and the major arc of the great circle. In each case let x be the arc length. rc length of }} 5 m x 808 } } 5 } 2πr 7π x < 1.6π rc length of }} 5 m x } } 2πr }} 7π x < 5.4π 7 heckpoint omplete the following exercises. 1. What is the sum of the angles of an equilateral triangle on a sphere? Find the distance between and. diameter 5 25, m The distances are 3.125π and π. Homework 3. Find the distance between and. diameter 5 30, m The distances are 10π and 20π. opyright Holt McDougal. ll rights reserved Focus on Reasoning Geometry Notetaking Guide 311
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