Each side of a regular septagon (7 sides) is 10cm long. What is the radius of the inscribed circle?

Question

 I don't want it done using a formula. I want to learn how to do it using cos, tan, sin or sin(a)/A = sin(b)/B, sin(b)/B=sin(c)/C.

Answer 1

Your typing would have been easier had you typed Law of Sines instead of 
sin(a)/A = sin(b)/B, sin(b)/B=sin(c)/C."

 

I believe the conventional name of a 7-sided polygon is heptagon. 

 

Draw a picture, using one of the triangles whose vertices are the center of the circumscribing circle, and the two endpoints of one of the sides. Due to regularity, you can compute the central angle, 360^o/7. Draw the perpendicular from the center to the side of the selected triangle. Now you will know all three of its angles, and can use simple trigonometry to calculate the length of the apothem, which will lead you to triangle's area...

Answer 2

draw the regular heptagon(septagon)

from the center of the inscribed circle, draw two line segments to two adjacent vertices of the heptagon

this forms an isosceles triangle (there are actually 7 such triangles but you need only one)

find the central angle by dividing 360º by 7

360/7=51.428571º

using the law of cosines (look it up if you don't know it)

call the equal sides of the triangle you formed x and x

10^2=x^2+x^2-2*x*x*cos(51.428571º)

100=2x^2-2x^2(0.6234898)

100=2x^2-x^2(1.246978)

100=0.753022x^2

100/0.753022=x^2

132.798=x^2

take the square root of both sides

x=√132.798

x=11.5238cm is the length of each side of the isosceles triangle

draw the altitude or apothem of the triangle which will be the radius of the circle

this forms two right triangles

this also divides the base side of the triangle into two equal segments, 5 cm each

use the Pythagorean Theorem to find the altitude or apothem or radius, whichever you want to call it

the hypotenuse is one of the equal sides, half the side of the heptagon is one leg and the apothem/altitude/radius is the other leg

11.5238^2=5^2+R^2 where R=radius/altitude/apothem

132.798=25+R^2

132.798-25=R^2

107.798=R^2

take the square root of both sides

R=√107.798

R=10.38258cm is the radius of the circle