If the length plus the width of rectangle ABCR is 8, then find perimeter of circle given rectangle is in a circle

Question

In the figure, arc SBT is one quarter of a circle with center R and radius $6$. If the length plus the width of rectangle ABCR is $8$, then find the perimeter of the shaded region.

Background: This is a problem given in the Archives book containing previous year SAT question

My solution: Here is the figure

Given $r=6$

arc $SBT=\dfrac{2\pi r}{4} \implies 3\pi$

Now calculating perimeter of remainder sides

$ CA+AS+CT $

Trying to relate them with radius

$ (SR-AR)+AC+(RT-RC) $

$ SR+AC+RT-(AR+RC)$

Since ABCR is a rectangle, $AR+CR=AR+RC=8$ cm

So the expression reduces to

$ 6+6+6-8=10 $

So finding perimeter

$$ 10+3\pi $$

Is my solution correct or is there any mistake? Also is there any alternate solution?

Answer 1

Your solution is correct. Here is another approach:

The perimeter of region is:

$(AS)+ SBT+CT+(AC=6)=(6-AR)+ 3\pi+(6-RC)+6=12-(AR+RC=8)+6+3\pi=10+3\pi$