How do we calculate the area of an octagon?

it is very easy as long as the octagon is a regular polygon.A=1/2(AP) A stands for apothem and P stands for perimeter.

1) Do you mean a *regular* octagon?


2) One idea is to inscribe the regular octagon in a square.
"The area may also be found this way:
A = S² - B²
Where S is the span of the octagon, or the second shortest diagonal; and B is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside (making sure that four of the eight sides touch the four sides of the square) and then taking the corner triangles (these are 45-45-90 triangles) and placing them with right angles pointed inward, forming a square. The edges of this square are each the length of the base."
Source and further information::
http://en.wikipedia.org/wiki/Octagon

Further information::
http://mathcentral.uregina.ca/QQ/database/QQ.09.01/laurie2.html

With a calculator

First you need to assume that the size of one side of the octagon is a.
Then on the octagon you can then easily draw two vertical and two horizontal lines which will divide the area into 9 pieces. The center piece is a square with side a. Its area is a x a.

This leaves us with 4 more rectangles and 4 triangles.

4 triangles are right angled triangles whose hypotenuse is a and whose other two sides are equal.

To find the area of each triangle, we need to find the length of its side. Using the Pythagoras theorem,
(x^2 + y^2 = a^2), and realizing that x and y (the otehr two sides of the triangle) are equal, we get
x = a / sqroot2
(a divided by the square root of 2).

The area of a triangle is b*h/2 (one half times base times height). Since base and height are both x, or a/sqrt2, the area of each triangle is a^2/4
(a squared over four).
The four triangles combined therefore have an area of a^2 (a squared)

Back to the 4 rectangles. They each have one side a and the other side shared with a triangle, therefore a/sqrt2.
The area of each rectangle is a squared over the square root of two (a^2/sqrt2).

So our total area is a^2 + a^2 + 4(a^2/sqrt2)

Taking a^2 common it equals (2+4/sqrt2)*a^2

so approximately the area is 4.8284*a^2.


Hope this helps

There is a similar online calculator here:

http://www.utilitiesman.com/measures/area/octagon