In the given fig., ABC is an equilateral triangle inscribed in a circle of radius 4 cm. Find the area of the shaded region.

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- Feb 28th 2010, 07:02 AMsnigdhaequilateral triangle inscribed in a circle..
In the given fig., ABC is an equilateral triangle inscribed in a circle of radius 4 cm. Find the area of the shaded region.

- Feb 28th 2010, 11:16 AMmasters
Hi snigdha,

The area of the circle is $\displaystyle A_c=\pi r^2=16 \pi$

We need to find the area of the triangle, subtract that area from the area of the circle and then divide it by 3.

Find the area of the equilateral triangle:

Draw the radius from O to B.

Draw the perpendicular bisector of BC through O intersecting BC at X.

Solve the right triangle we just made.

Hypotenuse =**4**

Since it's a 30-60-90 right triangle, angle OBX = 30 degrees and angle BOX = 60 degrees.

With a little trig or 30-60-90 rules, we determine that**OX = 2**, and BX = $\displaystyle 2\sqrt{3}$.

Since triangle OBC is isosceles,**BX = CX**.

The base of the equilateral triangle ABC is $\displaystyle 4\sqrt{3}$.

All we need now is the height of the equilateral triangle.

We just add the radius**4**to**OX**and get**6**.

The area of the inscribed equilateral triangle is $\displaystyle A_t=\frac{1}{2}bh=\frac{1}{2}(4\sqrt{3})(6)=12\sqr t{3}$.

Now you have both pieces I talked about in the beginning.

You may finish up now.

Turn the lights out when you leave. - Feb 28th 2010, 12:33 PMPlato
You can also look at this: circular segement.

- Feb 28th 2010, 01:40 PMmasters
- Feb 28th 2010, 09:52 PMsnigdha
Well.....i found masters' solution better and way simpler....!

thanks a lot!!