# Thread: Problem Solving #2

1. ## Problem Solving #2

Take a circle of radius 5, and mark 12 points that are equidistant around the circle.
Draw a line segment connecting each of these points with its adjacent points.
What is the area of the space between the circle and the lines?

2. Unclear. Diagram?

3. Hello, matgrl!

Take a circle of radius 5, and mark 12 points that are equidistant around the circle.
Draw a line segment connecting each of these points with its adjacent points.
What is the area of the space between the circle and the lines?

We have a circle with an inscribed regular dodecagon (12-sided polygon).

The area of the circle is: . $\pi r^2 \:=\:25\pi\text{ units}^2.$

The dodecagon is composed of 12 congruent isosceles triangles
. . with two sides of length 5 and the included angle $30^o.$

The area of one triangle is: . $\frac{1}{2}(5^2)\sin30^o \:=\:\frac{25}{4}\text{ units}^2.$

The area of the dodecagon is: . $12 \times \frac{25}{4} \:=\:75\text{ units}^2.$

Therefore, the difference is: . $25\pi - 75 \;\approx\;3.54\text{ units}^2.$

4. Dont have one. This is the only information I was given.

5. Thank you very much Soroban. I have a few questions that you might be able to clear up for me.

The area of a triangle is A = 1/2 b * h

-Why are you then using 5^2?

Also how did you know to use sin 30 degress =25/4 where did your 4 come from?

Thank you for your help...this is wonderful!

6. Do you "see" that 12 isosceles triangles with equal sides = 5 and equal angles = 75 degrees
are created? If you do, then do you not know how to calculate the area of one of these triangles?