An equilateral triangle of side 12 is inscribed in a circle

Question

An equilateral triangle of side 12 is inscribed in a circle, what is the area of the circle?

MBAhereIcome · Accepted Answer

remember these formulae for equilateral triangles and you'll save important seconds.
s = side

area of equilateral triangle =  \frac{s^2}{4}\sqrt{3} , height =  \frac{s}{2}\sqrt{3}

radius of circumscribe circle =  \frac{s}{\sqrt{3}} , radius of circumscribe circle =  \frac{s}{2\sqrt{3}}

radius of circle =  \frac{12}{\sqrt{3}} = 4\sqrt{3}  .... s=12

area of circle =  48\pi

arkadiyua · Answer

Thanks for the morning warmup!

C - 48P

The height of the triangle equals 63√63 since the sides are 6 and 12 (1:2: \sqrt{3} ratio). So the area of the triangle is (63√∗12)/2=363√(63∗12)/2=363

Since the triangle is equilateral, we can get 3 equal triangles with area of each 363√/2=123√363/2=123. From this we can get the heights of the smaller triangles: 123√∗2/12=23√123∗2/12=23. Thus, the radius is 63√−23√=43√63−23=43. The area of the cirle is (23√)2∗P=48P

yangsta8 · Answer

Was racking my brain on this one. Nice solution arkadiyua.

srivas · Answer

Soln. I too go with the Ans C - 48pi

konayuki · Answer

Hi arkadiyua.

I need your help. What do you mean by "we can get 3 equal triangles with area of each..."? I am lost in how to get this three triangles.

Since the triangle is equilateral, we can get 3 equal triangles with area of each  36\sqrt{3}/2=12\sqrt{3} .

yangsta8 · Answer

If you draw a line from the centre of the circle to each of the triangle vertices you will see that the triangle is divided into 3 equal triangles. In fact if you just draw a triangle… and draw a line from the centre to each of the vertices that will have the same result.

Helps to draw and visualise it.

konayuki · Answer

Thank you! I got it now.

praveenmittal95605 · Answer

C - 48P

The height of the triangle equals 63√63 since the sides are 6 and 12 (1:2: \sqrt{3} ratio). So the area of the triangle is (63√∗12)/2=363√(63∗12)/2=363

Since the triangle is equilateral, we can get 3 equal triangles with area of each 363√/2=123√363/2=123. From this we can get the heights of the smaller triangles: 123√∗2/12=23√123∗2/12=23. Thus, the radius is 63√−23√=43√63−23=43. The area of the cirle is (23√)2∗P=48P

Jaya6 · Answer

Hello,
if the area is equal for 3 triangles, shouldn't it be divided by 3 instead of 2?
also can you explain how did you get the value of height if smaller triangles if you knew the area 12root3?

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An equilateral triangle of side 12 is inscribed in a circle

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An equilateral triangle of side 12 is inscribed in a circle

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