Diagonals of the rhombus 12 and 9. Find the ratio of the area of the inscribed circle to the area of the rhombus?
October 14, 2021
The area of a rhombus is half the product of its diagonals: Srombus = d1 * d2 / 2 = 12 * 9/2 = 54.
The radius of the inscribed circle can be determined by the formula: r = d1 * d2 / 4 * a, where a is the side of the rhombus.
The side of the rhombus can be found from a right-angled triangle formed by the side of the rhombus and halves of the diagonals of the rhombus, as the square root of the sum of squares of half the lengths of the diagonals: a = √ ((d1 / 2) ^ 2 + (d2 / 2) ^ 2) = √ (( 9/2) ^ 2 + (12/2) ^ 2) = √ (81/4 + 144/4) = √ (225/4) = 15/2 = 7.5.
Find the radius of the inscribed circle: r = d1 * d2 / 4 * a = 9 * 12/4 * 7.5 = 108/30 = 3.6.
Area of the inscribed circle: Scircle = πr ^ 2 = 12.96π.
The ratio of the area of the inscribed circle to the area of the rhombus: Scircle / Srombus = 12.96π / 54 ≈ 0.754.
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