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Question

A craftsman makes square tiles of 20 cm on the side and paints them white and grey as shown in the figure, where all semicircles are of the same size.

craftsman must buy gray paint for that the craftsman needs to know the area of the gray region on each tile.
What is the value of the area, in cm2, grey on each tile?

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Answer 1 · 2017-12-05T17:33:54

$800$ $800$ [ ${cm}^{2}$ $"cm"^2$ ]

Let the square side be $2 r$ $2r$

The semi-lobe area is given by

$π \frac{r^{2}}{4} - \frac{r^{2}}{2} = (\frac{π}{4} - \frac{1}{2}) r^{2}$ $pi r^2/4-r^2/2 = (pi/4-1/2)r^2$

and the lobe area is

$2 (\frac{π}{4} - \frac{1}{2}) r^{2}$ $2(pi/4-1/2)r^2$

so the grey area is

$4 π r^{2} - 8 (2 (\frac{π}{4} - \frac{1}{2}) r^{2}) = 8 r^{2}$ $4pi r^2-8(2(pi/4-1/2)r^2) = 8r^2$

now for $r = 10$ $r=10$ [cm] we have

gray area $= 800$ $= 800$ [ ${cm}^{2}$ $"cm"^2$ ]