ABC is an equilateral triangle inscribed in a circle of radius 4 cm. Find the area of shaded reigon. Leave the answer in pie or surds form
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which area is shaded ? is it outside the ABC and the inside the circle?
Equilateral triangle : side = 4 cm. Each angle = 60 degrees.
Altitude = height of triangle = AB * sin 60 = 4 *√3/2 = 2 √3 cm
Area triangle ABC = 1/2 * 4 * 2√3 = 4 √3 cm²
Area of circle = π r² = π (4/√3)² = 16π /3 cm² as radius of circum circle = AB / √3
Shaded area = 16π/3 - 4√3 cm²
DO you need to derive the radius of circumcircle of ABC?
Let center of circle be O. Md point of AB = D. AO bisects angle A. => in triangle OAB, angle OAB = 30 deg. Draw a perpendicular from O onto AB intersecting AB at D.
Cos angle A/2 = AD / AO = AB/2 / r = AB / 2r
cos 30 = √3 / 2 = AB / 2r => r = AB / √3
Equilateral triangle : side = 4 cm. Each angle = 60 degrees.
Altitude = height of triangle = AB * sin 60 = 4 *√3/2 = 2 √3 cm
Area triangle ABC = 1/2 * 4 * 2√3 = 4 √3 cm²
Area of circle = π r² = π (4/√3)² = 16π /3 cm² as radius of circum circle = AB / √3
Shaded area = 16π/3 - 4√3 cm²
DO you need to derive the radius of circumcircle of ABC?
Let center of circle be O. Md point of AB = D. AO bisects angle A. => in triangle OAB, angle OAB = 30 deg. Draw a perpendicular from O onto AB intersecting AB at D.
Cos angle A/2 = AD / AO = AB/2 / r = AB / 2r
cos 30 = √3 / 2 = AB / 2r => r = AB / √3
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Answer:
16π/3 - 4√3 cm²
Step-by-step explanation:
height of triangle = AB×sin 60 = 4×√3/2 = 2 √3 cm
Area triangle ABC = 1/2 × 4 ×2√3 = 4 √3 cm²
Area of circle = π r² = π (4/√3)² = 16π /3 cm² as radius of circumference of circle = AB / √3
Area of the Shaded area = 16π/3 - 4√3 cm²
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