a regular hexagon is inscribed in a circle with a radius of 2 inches . what is the area of shaded region?

Question

express your answer as a decimal to the nearest hundreths

Answer 1

Hi Abhinav from Fort Smith, AR...hope you had a great week in school.

Great Geometry word problem. Well...I'll help you understand this, give you some hints on working to solve these kinds of problems. I won't answer it for you, that would be to easy. I will help you understand on how to set this up so if similar problems come up, like during a test or the ACTs or SATs, you'll be a pro at answering them.

Let's begin with what we know, you have a circle, which has 360°...right?

You also know the radius of the circle, which is  2 inches.

And you also know you have a regular hexagon, which means all the sides are equal and because the problem states it fits inside the circle, from each of the vertices of the regular hexagon to the center of the circle, that side is 2 inches long. We can then also say, each side is 2 inches long, making a triangle a specific kind of triangle...an isosceles triangle. 

Now, since there are how many sides to the hexagon...you should know this...6 sides, making 6 lines formed by the radius which make each of the legs of the center angle. You should be able to calculate the angle created by each of those legs to the center of the circle. If you are getting confused, using a compass, draw a circle, and draw a hexagon inside the circle. Connect the vertices of each of the sides of the hexagon through the center. Each of those triangles you created has an angle of what?

Your answer should be 60°. 

Now, using some extrapolation, since the triangles are isosceles triangles and the angle created by the lines through the center is 60° what are the remaining angles created by the base of the triangles, what is the sides of the hexagon. Okay, so now, I'm giving you way too much, you should be able to figure out the rest...

Quick review...you have 6 triangles inside a circle with a radius of 2 inches.

The sides of the isosceles triangle, which means all the sides, EXCEPT the base are ALL equal.

And the angle created by the lines through the center create an angle of 60.

What would be the angles of the remaining angles at the base of the isosceles triangle, and from there you can calculate the length of the base of each of the isosceles triangles.

Let me know if you have further questions.