Math Angle Measures Key

Angle Measures: Applications Length of an Arc: part of the circumference S = (θ/2π) * 2πr = θr where θ is in radians Example 1: a. Find the length of an arc of a circle with radius 21 m that subtends a central angle of 15°. S = (15°/180°) * π * 21 = 5.50 m b. A central angle, θ, in a circle of radius 9 m is subtended by an arc of length 12 m. Find θ in radians. S = rθ 12 = 9θ θ = 1.3 radians Area of a Sector: part of the area A = (θ/2π) * πr² = 1/2 * r²θ where θ is in radians Example 2: Find the area of a sector with central angle 4° and diameter 90 cm. 4° * (π/180°) * 1/2 * (45)² = 70.69 cm² Example 3: If the area of a circle is 64 cm², find the area of a sector of this circle that subtends a central angle of π/4. 64 = 1/2 * r² * π π/4 is 1/8 of a circle 64 * 1/8 = 8 cm² Linear and Angular Speed * Suppose a point moves along a circle. Let s be the distance the point travels in time t. **LINEAR SPEED:**   r = distance/time = arc length/time = s/t **ANGULAR SPEED:** r = distance/time = angle/time = θ/t * angular speed * times radius = linear speed EQUIVALENT RATIOS: 1 revolution = 2π radians = 360° = Circumference (2πr units, πd units) **Example 4:** A ceiling fan with 16-inch blades rotates at 45 rpm. a. Find angular speed in rad/sec 45 rev/min * 2π rad/1 rev = 90π rad/min = 4.71 rad/sec b. Find linear speed in in/sec r = 16 in 45 rev/min * 1 min/60 sec * 2π(16) in/1 rev = 75.4 in/sec **Example 5:** A woman is riding a bicycle whose wheels are 30 inches in diameter. If the wheels rotate at 150 rpm, find the speed at which she is traveling in mi/hr. r = 15 in 150 rev/min * 1 min/60 sec * 2π(15) in/1 rev * 1 ft/12 in * 1 mile/5280 ft * 60 min/1 hr = 13.4 mi/hr **Example 6:** The circular blade on a saw has a diameter of 7.25 inches and rotates at 4800 revolutions per minute. Find the linear speed of the saw teeth (in feet per second) as they contact the wood being cut.   d = 7.25 in r = 3.625 in 4800 rev/min * 1 min/60 sec * 7.25π in/1 rev * 1 ft/12 in = 151.8 ft/sec Angle Measures: Applications Find the length of the arc s in the figure. S = rθ = 7 * 100° * (π/180°) = 12.21 units Find the length of the arc s in the figure. S = rθ = 5 * 140° * (π/180°) = 19.2 units Find the length of an arc that subtends a central angle of 45° in a circle of radius 10 m. S = rθ = 10 * 45° * (π/180°) = 7.85 m Find the length of an arc that subtends a central angle of 2 rad in a circle of radius 2 mi. S = rθ = 2 * 2 = 4 mi An arc of length 100 m subtends a central angle θ in a circle of radius 50 m. Find the measure of θ in radians and degrees. 100 = 50θ θ = 2 rad θ = 2 rad * (180°/π) ≈ 114.6° Find the radius of the circle if an arc of length 6 m on the circle subtends a central angle of π/6 rad. 6 = r * π/6 r = 36/π ≈ 11.5 m Find the area of a sector with central angle 1 rad in a circle of radius 10 m. A = 1/2 * r²θ = 1/2 * 10² * 1 = 50 m² Find the area of a sector with central angle 80° in a circle of radius 8 mi. A = 1/2 * r²θ = 1/2 * 8² * (80° * π/180°) ≈ 44.7 mi² The area of a sector with a central angle of 2 rad is 16 m². Find the radius of the circle. 16 = 1/2 * r² * 2 r² = 16 r = 4 m A sector of a circle of radius 24 m has an area of 288 m². Find the central angle of the sector in rad. 288 = 1/2 * 24² * θ θ = 1 rad The area of a circle is 72π cm². Find the area of a sector of this circle that subtends a central angle of π/6 rad. πr² = 72π r² = 72 r = √72 A = 1/2 * (√72)² * π/6 = 6 cm²