ABE is an equilateral tringle, BCDE is a square, and the are

Question

Untitled.png In the figure above, ABE is an equilateral triangle, BCDE is a square, and the area of the circle is 36π. What is the area of polygon ABCDE? A. 108 + 27 \sqrt{3} B. 108 + 36 \sqrt{3} C. 189 D. 216 + 81 \sqrt{3} E. 324 + 27 \sqrt{3}

caioguima · Answer

Given that ABE is an equilateral triangle, the Radius is equal to 2/3 of the triangle's height:

R = (2/3).L\sqrt{3} /2 = 6

Hence,  L = 6\sqrt{3}

The area of the region ABCDE is therefore:

Area = L^2 + (L^2).\sqrt{3}/4

= 108 + 27\sqrt{3}

ANSWER: No option available (maybe a typo on answer choice A)

holidayhero · Answer

https://i.ajdesigner.com/cdn/equilateral_triangle_circumscribed_circle_radius_r_equation.png

R = radius of circle
a = side of equilateral triangle (also side of square)

We know R = 6 because area of circle is 36π
Plug this into the formula above and you will see that a = 6\sqrt{3}

Let Area of Equilateral Triangle be represented by "K" 
 https://i.ajdesigner.com/cdn/equilateral_triangle_area_a_equation.png  
(sub in a = 6\sqrt{3} )
 K = 27\sqrt{3}

Let Area of Square be represented by "S" 
 S= a^2 
 S=(6\sqrt{3})^2 
 S=108

Total Area of Polygon ABCDE is
 108 + 27\sqrt{3}

pancakeFR · Answer

Hi, I don't understand where you got that from. Could you explain please ?

Thank you.

holidayhero · Answer

https://i.ajdesigner.com/cdn/equilateral_triangle_circumscribed_circle_radius_r_equation.png

Here's an explanation. Caioguima also explained it.

Let the side length of the equilateral triangle be  a . 
The circumradius of the equilateral triangle is equal to  2/3  of the altitude. The altitude is equal to  a\sqrt{3}/2  (Drawing the altitude creates two 30,60,90 triangles). The circumradius of this triangle is also the radius of the circle.

R =  (2/3) (a\sqrt{3}/2) 
R =  a\sqrt{3}/3

SpotlessMind · Answer

in the equilateral triangle, the circum-centre divides the height in the ratio 2:1 (smaller length towards the base)

and area of circle = pi x r*r hence r= 6

so 2/3 * (3^ 1/2)/2*a = 6 where a is side of triangle
hence solving for a we get a = 6* (3^0.5)

the required are = a*a + (3^0.5)/4 * a*a

= {108 + 27(3^0.5)}

pancakeFR · Answer

Thank you all for your answers. I get it now !

vishalrastogi · Answer

Hi Bunuel,

Will you please explain this ? I am not even close to understanding !

Thanks.

b2bt · Answer

This might not be the best approach. But it can do the trick if you don't know how to proceed with the sum
Area of triangle = √3/4 a^2
Area of square = a^2

Total area = area of triangle + area of square = √3/4 a^2 + a^2

If you see the options except C all have √3 part. It is very less likely for our answer to not have √3 part.
(this is a part where we might be wrong the the answer might have been approximated to near number, unless if GMAT doesn't do so without specifying)

Now the other options.
Lets see, if we have a + x√3 in answer choices then it must be true that x = 4a (Why? try playing with the total area)
This only true for option A. Voila!

Bunuel · Answer

https://gmatclub.com/forum/download/file.php?id=22496 In the figure above, ABE is an equilateral triangle, BCDE is a square, and the area of the circle is 36π. What is the area of polygon ABCDE? A. 108 + 27 \sqrt{3} B. 108 + 36 \sqrt{3} C. 189 D. 216 + 81 \sqrt{3} E. 324 + 27 \sqrt{3} The area of the circle is 36\pi --> \pi{R^2}=36\pi --> R=6. The radius of the circumscribed circle in equilateral triangle is R=side\frac{\sqrt{3}}{3}=6 --> side=6\sqrt{3} . The area of equilateral triangle = side^2*\frac{\sqrt{3}}{4}=27\sqrt{3} . The area of square = side^2=108 . The area of polygon ABCDE = 108+27\sqrt{3} . Answer: A. For more check Triangles chapter of Math Book here: math-triangles-87197.html Hope it helps.

unceldolan · Answer

Hi Bunuel,

I have several questions regarding this matter.

First: is this 
"The radius of the circumscribed circle in equilateral triangle is  R=side\frac{\sqrt{3}}{3}  "
a general rule? If a equilateral triangle is inscribed in a circle, the formula applies?

And how do you get from the first step to the second step?

R=side\frac{\sqrt{3}}{3}=6  -->  side=6\sqrt{3}

The way I see it, 6 = side * sqrt3 / 3 => 18/sqrt3 . What am I mssing?

Rest is clear.

Thank you

bazc · Answer

1)
Please refer to the Math resources section on geometry for relationships between circumradius, inradius, etc wrt equilateral triangles. It will be useful to learn these concepts so you can save time.

2)
18/sqrt3
Multiple both numerator and denominator with sqrt3,
= 6*sqrt3

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ABE is an equilateral tringle, BCDE is a square, and the are

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