11.1 Circumference and Arc Length Geometry EQ: How can you find the length of a circular arc?

Geometry 11.1 Circumference and Arc Length Goals Find the circumference of a circle. Find the length of an arc. Solve problems using circumference and arc length. February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length The circumference is the distance around a circle. February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Pi The ratio of the circumference to the diameter is called pi. Use the Greek letter for pi:  d C February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length More About Pi.   3.14159… The fraction 22/7 is close. 355/113 is closer. Current computer calculations have pi to over 5 trillion places. February 18, 2019 Geometry 11.1 Circumference and Arc Length

Pi to a few decimal places: 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837297804995105973173281609631859502445945534… February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Pi is Irrational  does not repeat.  does not terminate. February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length You can find the circumference if you know the diameter (d) or the radius (r) of the circle. February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Example Find the circumference of a circle with a radius of 8. This is the circumference “in terms of pi”. When computing C in decimal, use 3.14 or the pi button. February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length The ratio of the length of an arc to the circumference of a circle is equal to the ratio of the measure of the arc to 360. A B Recall: the measure of the arc is equal to the central angle that intercepts it. February 18, 2019 Geometry 11.1 Circumference and Arc Length

Example Find the length of AB. 5 80 L Write the proportion: Length of Arc Central Angle Circumference of circle February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Solve the proportion A B 5 80 6.98 L February 18, 2019 Geometry 11.1 Circumference and Arc Length

Your Turn Find the length of AB 8 25 x February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Example A B 10 x 25 The length of AB is 25 and the radius is 10. Find the mAB. February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Semicircle Half of a circle. Its measure is 180. Its length is one-half the circumference. February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Problem 1 An indoor track at a gym has the design below. What is the length of the entire track? 40 ft 100 ft February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Problem 1 Solution The ends make one circle. The diameter is 40 ft. C = (40) 40 ft 100 ft February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Problem 1 Solution Add 200 for the straight segments. Perimeter = 40 + 200 ft Or, 325.66 ft. 40 ft 100 ft February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Problem 2 Using a GPS, it is found that Tombstone is at 31.7 N latitude, and Show Low is at 34.3 N. Both are at 110 W longitude. If the radius of the earth is 3963 miles, how far apart are these two towns? 34.3 ? 31.7 February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Simplified 34.3 2.6 31.7 3963 mi Equator February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Problem 2 Solution Using a GPS, it is found that Tombstone is at 31.7 N latitude, and Show Low is at 34.3 N. Both are at 110 W longitude. If the radius of the earth is 3963 miles, how far apart are these two towns? Change in latitude 34.3 – 31.7 = 2.6 February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Problem 2 180 miles. How good is this? This is a good estimate. 180 160 120 80 40 February 18, 2019 Geometry 11.1 Circumference and Arc Length

Converting between Degrees to Radians The Radian Measure of a central angle can be thought of as the length of the associated arc. This concept will be explored more thoroughly in Trig.   EX: convert 30 degrees to radians.   February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Practice 1 The circumference of a circle is 200 ft. Find its radius. February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Practice 2 The radius of a circle is 16 in and central angle ACB measures 42. Find the length of arc AB. February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Practice 3 An arc of a circle has a length of 12 cm. Its central angle is 15. What is the radius of the circle? February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Practice 4 A circle has a diameter of 20 cm. An arc on the circle is 10 cm in length. Find the measure of the central angle. February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Practice 5 At an altitude of 237 miles, a space ship travels in orbit an angle of 2. How many miles is this? (Recall that the radius of the earth is 3963 mi.) February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Practice 5 At an altitude of 237 miles, a space ship travels in orbit an angle of 2. How many miles is this? (Recall that the radius of the earth is 3963 mi.) 237 3963 2 February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Practice 5 237 3963 2 February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Problem A string is tied snuggly around the Earth, from pole to pole. The radius of the Earth is approximately 3963 miles. 3963 mi February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Problem continued A second string is tied around the Earth so that is stands off the surface exactly 1 inch for the entire circumference. 1 inch 3963 mi February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Question How much longer is the second string than the first one? 1 inch 3963 mi Useful info: 1 mi = 5280 ft 1 ft = 12 in February 18, 2019 Geometry 11.1 Circumference and Arc Length

String Around the Earth Solution How much longer is the second string than the first one? 1 inch 3963 mi Useful info: 1 mi = 5280 ft 1 ft = 12 in February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Solution First, change 3963 mi to inches. 3963 mi  5280 ft/mi  12 in/ft = 25090560 in That’s the radius of the inner string. The radius of the outer string is 25090561 in February 18, 2019 Geometry 11.1 Circumference and Arc Length

Find both circumferences February 18, 2019 Geometry 11.1 Circumference and Arc Length

Now find the difference 157648644.2 - 157648637.9 6.3 inches That’s right! A string standing one inch off the surface of the Earth, all 24,900 miles of it, is only 6.3 inches longer than the snug-fitting one. Believe it or Not! February 18, 2019 Geometry 11.1 Circumference and Arc Length

Geometry 11.1 Circumference and Arc Length Summary The length of an arc on a circle is proportional to the measure of the arc and to the central angle. A semicircle has 180 and it’s length is half the circumference. February 18, 2019 Geometry 11.1 Circumference and Arc Length