Question

Given that the curves are formed from quarter circles, find the area of the shaded region. Give your answer in terms of T. 12m 12m​

Answer 1

the area of the shaded region:

`= 2(the \ area \ of \ a \ quarter \ circle) - (the \ area \ of \ the \ square)\\`

`= 2\Big((\pi 12^(2) )/(4)\Big) - 12^(2) = (144 \pi)/(2) - 144 = 72 \pi - 144\\`

`= 72 (\pi - 2) \ m^(2)`

Answer 2

72π-144 m²

Explanation:

You want the area of a shaded region consisting of the overlap of two quarter circles in a 12 m square.

Segments

If we draw a diagonal from upper left to lower right through the figure, the shaded area is divided into two 90° segments of a circle of radius 12 m.

The formula for the area of a segment is ...

A = 1/2r²(θ -sin(θ))

where θ is the measure of the central angle.

For θ = π/2 radians, this is ...

A = 1/2r²(π/2 -1) . . . . . half the shaded area

Shaded area

Then the whole shaded area is ...

2 × 1/2r²(π/2 -1) = (12 m)²(π/2 -1) = 72π -144 m²

The area of the shaded region is 72π -144 m².

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Additional comment

If we expand the shaded area formula, we get ...

A =1/2πr² -r²

This is recognizable as twice the area of a quarter circle, less the area of the square.

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