What is the area of a regular decagon with a radius of five?

Answer 1

If by radius you mean the distance from the center to the verticies, which is called the circumradius, the answer is

A = 125*sin(144º)
= 125*√(10-2√5) / 4
[which is the exact value after substituting for the sin(144º)]
≈ 73.47315654

[There are 10 triangles, each with:
height = 5*sin(72º) & ½*base = 5*cos(72º)
The area of each triangle is then:
25*sin(72º)cos(72º),
and the total area is:
250*sin(72º)cos(72º) = 125*sin(144º) ]

The link the answerer posted above will verify this. His answer is for a decagon with in inradius of 5, which seems counter to what you wanted.

Answer 2

A regular decagon can be considered to be that of 10 triangles each with two sides equal to the radius (R) and a base (b). The apex angle of each triangle is 36 degrees. The other two equal angles are each 1/2 x (90 - 36) degrees or 27 degrees. The base length is gotten from the relation: (b/R = Tan 36/2 degrees) The height (h) of each triangle is gotten from the relation: h^2 = R^2 - (b/2)^2 or h = the square root of [R^2 - (b/2)^2].

So the area (A) = 1/2 x (R x Tan 18 degrees) x {square root [R^2 - (b/2)^2]}

And the Area (D) of the decagon is 10 x A, or 5 x (5 x Tan 18 degrees) x {square root [R^2 - (b/2)^2]}.

D = 5(5 x Tan 18) x { Sq. rt. [25 - (2.5 x Tan 18)]}. Answer

I'll let you evaluate this equation.

Answer 3

Area is 73.15, i have almost the same answer as the 2nd on but i think because of the significant figures it changed a bit. my solution is different from the 2nd man who answered your question.

divide the circle by 10 (360 degrees divided by 10) you would get 36 degrees then you have 10 triangles that has 36 degrees on top then cut the triangle to half so that it would be a right triangle and then use cosine and sine to get the base and height. then use A=[1/2 BH] X 10, times it by ten because there is 10 triangles in the circle. then you have your area.

Answer 4

81.2