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, 08:02 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, 12:16 PMmasters
Hi snigdha,

The area of the circle is

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 = .

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

The base of the equilateral triangle ABC is .

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 .

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, 01:33 PMPlato
You can also look at this: circular segement.

- Feb 28th 2010, 02:40 PMmasters
- Feb 28th 2010, 10:52 PMsnigdha
Well.....i found masters' solution better and way simpler....!

thanks a lot!!