The diagram below shows two overlapping semicircles, with centres 1 and 2, inside a square.
The diameters of the semi-circles are equal to the length of the sides of the square. If the area of
the shaded region is equal to 8.9325 2
, find the perimeter of the shaded region.
Answers
Perimeter of the shaded region = 17.85
Step-by-step explanation:
Let say Side of Square = 2a
Area of Square = 2a * 2a = 4a²
Diameter - 2a
Radius = a
arcs intersect at 90°
Area of un shaded region = 2 arcs with 90° + a square of side a
= 2 * (90/360) π a² + a²
= π a²/2 + a²
Area with shaded region = 4a² - (π a²/2 + a²)
= 3a² - π a²/2
3a² - π a²/2 = 8.9325 2
=> a² = 6.25
=> a = 2.5
perimeter of the shaded region = 2a + 2a ( two sides of square) + 2 * arcs at 90 deg
= 4a + 2 * (1/4) 2πa
= 4 * 2.5 + 3.14 * 2.5
= 10 + 7.85
= 17.85
perimeter of the shaded region = 17.85
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