About Cynthia Watanabe Expertise I have taught high school mathematics for fourteen years. I can help students develop proofs, solve problems which require some algebra skills. Some topics that you may wish to explore include points, lines, planes, segments, rays, angles, polygons, solids, measurement, area, and volume.
Experience I have taught high school mathematics fourteen years.
Education/Credentials B.S. Secondary Education with a concentration in Mathematics. Master of Library and Information Science with a concentration in Archiving.
Question What is the radius of a circle which is inscribed into a pentagon with a side of 12?
Answer Indre,
The picture that you describe shows a circle inside of a pentagon. Now I am assuming that the pentagon is regular. That means that all the sides of the pentagon are = and all the angles of the pentagon are =.
Now put a dot at the center of your circle and draw the apothem of the pentagon. Remember, the apothem is the perpendicular segment from the center to one of the sides of the pentagon.
Now draw a segment from the center to one of the vertices of the pentagon next to the segment you used for the apothem. This makes a right triangle.
Did you notice that the apothem of the pentagon is also the radius of the circle. Yeah!
Since one of the sides is 12 units long then the short side of the right triangle that you just made is 6 units long.
What about the angle opposite the short side? To find that angle, we take 360 degrees and divide by 5(number of sides of polygon). The answer is 72 degrees for the central angle. The right angle that you created is half of the central angle so 72/2 = 36 degrees. The angle across from the side that you labeled as 6 is 36 degrees.
Since this right triangle is not a special right triangle, then we have to use trigonometry to find the length of the apothem.
Since 6 is opposite the 36 degree angle and the apothem is adjacent to it, the best trig. function to use is the tangent function. The equation is tan 36 = 6/a. When you solve this equation, you will find the apothem to be 8.26.
Since the apothem is the radius, then the radius is 8.26 units long.
Here's a website that is related(but not exactly like your question) to what you are doing and it has a nice picture. : www.tasonline.co.za/toolbox/area/polygon.htm
Let me know if you have any other questions. Take care, Cynthia
