Find the length of the perimeter of a regular decagon which surrounds a circle of radius 12 cm.
Answers
Step-by-step explanation:
perimeter of radius=1/2×r
=1/2×12
=6cm
Perimeter of Decagon = 60 (√5 - 1) = 74.16 cm
Step-by-step explanation:
decagon has 10 sides
Hence angle between two consecutive sides with center = 360°/10 = 36°
A triangle is formed where two sides are radius with angle between them = 36° and third side is Side of Decagon
=> (Side of Decagon)² = Radius² + RAdius² - 2 *Radius * Radius Cos36°
=> (Side of Decagon)² = 12² + 12 ² - 2 * 12² Cos36°
=> (Side of Decagon)² = 12² * 2 ( 1 - Cos36°)
=> (Side of Decagon)² = 12² * 2 ( 1 - Cos36°)
Cos36° = (√5 + 1)/4
=> (Side of Decagon)² = 12² * 2 ( 1 - (√5 + 1)/4)
=> (Side of Decagon)² = 12² * 2(3 - √5)/4
=> (Side of Decagon)² = 12² * (3 - √5)/2
(3 - √5)/2 = 3/2 - √5/2 = 5/4 + 1/4 - √5/2
= (√5 / 2)² + (1 /2)² - 2(√5/ 2)( 1/2)
= ((√5 - 1)/2 )²
=> (Side of Decagon)² = 12² *((√5 - 1)/2 )²
=> Side of Decagon = 12 * (√5 - 1)/2
=> Side of Decagon = 6 * (√5 - 1)
Perimeter of Decagon = 10 * 6 * (√5 - 1) = 60 (√5 - 1) = 74.16 cm
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