ABCD is a square with side a. In centres A,B,C and D four circles are drawn such that each circle touches externally two of the three remaining circles. Let δ be the area interior of the square and exterior of the circles. What is the value of δ?
δ = Shaded area δ = Area of a square - (4 x ¼ area of a circle) = a² - (area of a circle) = a² - πr² but r = ᵃ⁄₂ = a² - π(ᵃ⁄₂)² = a² - ((πa²) ÷ 4) = (4a² - πa²) ÷ 4 = a² [(4 - π) ÷ 4]
ABCD is a square with side a. In centres A,B,C and D four circles are drawn such that each circle touches externally two of the three remaining circles. Let δ be the area interior of the square and exterior of the circles. What is the value of δ?
δ = Shaded area δ = Area of a square - (4 x ¼ area of a circle) = a² - (area of a circle) = a² - πr² but r = ᵃ⁄₂ = a² - π(ᵃ⁄₂)² = a² - ((πa²) ÷ 4) = (4a² - πa²) ÷ 4 = a² [(4 - π) ÷ 4]
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