A few days ago
Anonymous

How do you find the area of and n-gon? read more for details?

I have to write an equation to find the area of an n sided polygon using only two variables, where R is the radius and N is the number of sides….please help! include steps if you can!!!! You can use more varibles while your working on the equation, but in the end you can only have N and R. THANKS!

Top 3 Answers
A few days ago
hayharbr

Favorite Answer

The radius is a segment going from the center to a vertex. The formula for area uses the apothem, a segment going from the center to the midpoint of a side. The apothem is also the altitude of one of the N isosceles triangles formed by drawing all the radii, and the length of a side is the base of these triangles. The apothem a, radius R , and half a side h make a right triangle. You can use trigonometry to find h. The angle at the center will be 360/2N or 180/N since there will be twice as many of these angles as there are sides.

h will be opposite leg and R hypotenuse so use sine: h = R sine (180/N). That makes the length of a side 2h or 2R sin(180/N)

a will be adjacent leg so use cosine: a = R cos(180/N)

This makes the area of each isosceles triangle

be 1/2 (2R sin(180/n)(R cos 180/N), or

R^2 sin(180/N)(cos 180/N) and there are N triangles so the area is

N R^2 sin(180/N)(cos 180/N)

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A few days ago
apatel1488
There’s a formula i remember for that, which is

Area = 1/2 * apothem * perimeter

The apothem of any N-gon is a straight line drawn from the center of the figure to the middle part of any side (NOT the vertex).

We also have to know one more equation:

[180 (n-2)] / n, which gives a single interior angle of each N-gon.

Ok, so draw a picture out, let’s say of a triangle.

Draw the radius out to one of the angles (bisect it) and then draw the apothem. You now have a small right triangle.

Now solve for the other leg using sine and cosine and the previous equation.

cos ( 1/2 (180) (n-2) / n) = side / radius

side = radius cos (90 (n-2) / n)

you can simplify this to

side = radius * sin (180/n)

Now double this number: 2 * radius * sin (180/n)

Now multiply it by n to get the perimeter =

2N * radius * sin(180 / n)

—–

Now solve for the apothem:

sin ( 1/2 * 180(n-2)/n) = apothem / radius

apothem = sin (90 * (n-2) / n) * radius

apothem = cos (180 / n) * radius

FINALLY…

plug it back in to the equation:

Area = 1/2 apothem * perimeter

A = 1/2 * 2 * sin (180/N) * cos (180/n) * NR^2

You can clean this up a bit

A = 1/2 * sin (360/N) * NR^2

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A few days ago
adenine
if you dont know it you need to go to your teacher and aks them before you have to turn in the hw becasue you NEED to know this stuff! you’ll see it many many more times its complicated but if you understan the consept you can get every problem.

do you have a book? look in there they should have a formula and examples as to how to do it. or go to mathhelp.com type in you book, author and stuff and they will have solved problems look for the chapter and such.

n-8

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