The corners of an equilateral triangle are rounded to form semi-circle

Question

The corners of an equilateral triangle are rounded to form semi-circles of radius 1. If the length of a side of the triangle before rounding is 10, what is the area of the resulting figure?

A.  25\sqrt{3} + 3

B.  25\sqrt{3} + \frac{3\pi}{2}

C.  25\sqrt{3}

D.  22\sqrt{3} + 3

E.  22\sqrt{3} + \frac{3\pi}{2}

GMATinsight · Answer

Area of the resulting figure = Area of Bigger equilateral - (Area cut out)

Area cut out from one corner = (1/3)*(Area of Equilateral triangle with side 2√3 - Area of Circle with radius 1) .

Area cut out from one corner  = (1/3)*  . 
Total Area cut out  = 3*  = 3√3 - π

Area of the resulting figure =  (√3/4)* 10)^2 - (3√3 - π) = 22√3 +π

I think answer must be option E but highlighted part is still a bit confusing. Taking semicircle out from corners will involve different calculations.

manjot123 · Answer

got the answer correct but took 3 mins first find the area of remaining figure = root3/4 10*10 - 3*root3/4*2*2( this is the area of equi triangle formed by making a semi circle and then add area of semi circle to remaining fig (pi*1^2/2 )*3

manjot123 · Answer

GMATNinja  
Any better way?

thoomba · Answer

Try to work out the sides of the big triangles and small triangle (pyto theorem) first.

(a) Area of the big equilateral triangle = (10/2) * 5 (3^(1/2)) = 25(3^(1/2))
(b) Area of 1 semi-circle = π (1^2) /2 = π/2
(c) Area of 1 small equilateral triangle= (3^(1/2))
(d) Area of 3 cut-out parts= (c)- (b) = 3* = 3  = 3(3^(1/2)) - 3π/2
Answer = (d) - (a) = 25(3^(1/2)) -   = 22(3^(1/2))+ (3π/2)------ (E)

Fdambro294 · Answer

Start with the Equilateral Triangle of side 10 and move through step by step.

First, find the area of the equilateral triangle before we do anything ——> (10)^2 * sqrt(3) * (1/4) =

25 * sqrt(3)

Next, we need to round down each edge and make a semi-circle with radius 1 in its place (diameter of 2)

Start with just one vertex.

For the semi-circle to sit on top of the triangle, the Diameter across the semi-circle must be parallel to the base side opposite the vertex.

With a Diameter of 2, that means we are cutting of a similar equilateral triangle with side 2 from the vertex.

We have to do this for all 3 of the vertices. So let’s find the area of the 3 removed edges/triangles.

(3) * (2)^2 * sqrt(3) * (1/4) =

3 * sqrt(3) ———-> and this is being removed from the equilateral triangle’s full area.

25 * sqrt(3) - 3 * sqrt(3) =

22 * sqrt(3)

Last, in place of the 3 similar equilateral triangles that we removed from each of the 3 vertices, we are installing 3 semi-circles with radius 1.

So, we need to add the area of these 3 semi-circles with radius 1 to the result above to get the final answer

Each semi-circle’s area:

(1/2) * (1)^2 * (pi) = (pi) * (1/2)

And we are putting 3 where each vertex used to be.

(3) * (1/2) * (pi) ———-> finally add this to the result obtained above and you get

Answer E

22 * sqrt(3) + (3/2) (pi)

Posted from my mobile device

bumpbot · Answer

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The corners of an equilateral triangle are rounded to form semi-circle

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The corners of an equilateral triangle are rounded to form semi-circle

Prep Toolkit

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