A circle is inscribed in a rhombus with a side length of 169 and one diagonal of length 130. What is the area of the region inside the rhombus but outside the circle? Express your answer to the nearest whole number.

 May 23, 2020

The diagonals of a rhombus are perpendicular to each other.


If we draw the rhombus and its two diagonals, we get four congruent right triangles.

Looking at one of these right triangles, we can see that the hypotenuse is 169 (the side of the rhombus) and one of its sides is 65 (one-half of the one diagonal0.

I'm going to use the Pythagorean Theorem to find the other half-diagonal:

  c2  =  a2 + b2   --->   1692  =  652 + b2   --->   b  =  156   --->  so the other diagonal is  312.


Now that I know the length of both diagonals, I can find the area of the rhombus:

  A  =  ½·diagonal1·diagonal2   --->   A  =  ½·130·312  =  20280


Since all four triangles are congruent, each triangle has an area of  20280 / 4  =  5070.


I'm going to use this area to find the radius of the inscribed circle.

The center of the circle is the center of the rhombus. 

The radius of the circle is the height of each of the triangles (the base will be a side of the rhombus).

  A  =  ½·base·height   --->   5070  =  ½·169·height  --->   height  =  60


The radius of the circle is 60.

Find the area of the circle and subtract from the area of the rhombus.

 May 23, 2020

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