Question

Find the area of the shaded portion in the equilateral triangle with sides 6. Show all work for full credit. (Hint: Assume that the central point of each arc is its corresponding vertex.) 

Answer 1

So... if you notice the picture below

each circle, has their central angle at the vertex of the triangle
that simply means, 3 circles with a radius of 3, overlapping the triangle

now, the area in the middle, the shaded one, will be, the whole area of the triangle MINUS those 3 circle sectors

hmmmm each sector has 60°, that means, all three of them will then be 60+60+60 or 180°, so the area of those three sectors, can be combined into a 180° sector, well, hell, 180° is really half a circle

so.... the area of those three sectors of 60° each, all three combined, is the same area of half a circle with a radius of 3

so    $\bf \textit{area of an equilateral triangle}\\\\ A=\cfrac{s^2\sqrt{3}}{4}\qquad s=\textit{length of one side}\\\\ -----------------------------\\\\ \textit{area of a circle}\\\\ A=\pi r^2\qquad r=radius\\\\ \textit{area of half a circle}\\\\ A=\cfrac{\pi r^2}{2}\\\\ -----------------------------$

$\bf \textit{now, let us use the side of 6, and radius of 3} \\\\\\ \begin{array}{clclll} \cfrac{6^2\sqrt{3}}{4}&-&\cfrac{\pi 3^2}{2}\\ \uparrow &&\uparrow \\ triangle's&&semi-circle's \end{array}\impliedby \textit{area of shaded area}\\\\ -----------------------------\\\\ \boxed{\cfrac{36\sqrt{3}}{4}-\cfrac{9\pi }{2}}$

you can, add the fractions if you want, or leave them like that, or get their difference by using their decimal format

Answer 2

Answer:

9 sqrt 3 - 4.5 pi

Step-by-step explanation:

So, first off, you need to find the area of the whole triangle. Since this is an equilateral triangle, the formula for it would be:  

A = 1/4 * s^2 * sqrt 3 ----> A = 1/4 * 6^2 * sqrt 3 ----> A = 9 sqrt 3. So the area of the whole triangle is 9 sqrt 3.

Next, imagine that there are three circles that are part of the sectors (I am using the same picture as jdoe0001 attached). So to find the area of each sector, you would do 60o (o = degree sign) divided by 360o --> 60/360. You should get 1/6. Then you would multiply this by the area of the circle, which you find by doing the following: A = pi * r^2 ----> A = 9 pi. So when you would multiply 1/6 by the area of the circle, you would get 1.5 pi. Since this is just one sector, you need to multiply 1.5 pi by 3, because there are 3 sectors. You should get 4.5 pi.

Finally, to get the area of the shaded portion, you would simply do this: the area of the whole triangle minus the area of the sectors ----> 9 sqrt 3 - 4.5 pi. This is the exact answer. The approximate answer would be 1.45.

Hope this helps! :)