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?
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: .$\displaystyle \pi r^2 \:=\:25\pi\text{ units}^2.$
The dodecagon is composed of 12 congruent isosceles triangles
. . with two sides of length 5 and the included angle $\displaystyle 30^o.$
The area of one triangle is: .$\displaystyle \frac{1}{2}(5^2)\sin30^o \:=\:\frac{25}{4}\text{ units}^2.$
The area of the dodecagon is: .$\displaystyle 12 \times \frac{25}{4} \:=\:75\text{ units}^2.$
Therefore, the difference is: .$\displaystyle 25\pi - 75 \;\approx\;3.54\text{ units}^2.$
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!