Find area formula for a dodecagon?

Question

What is the formula to find the area of a regular dodecagon?

THANKS

Answer 1

The formula for the area of a regular dodecagon, given that s is the length of one side, is given as follows:

A = 3(2 + √3) s².

Here’s why:

First, picture a twelve-sided figure of equal sides and angles, either in your mind or in an actual drawing. Assume, for the moment, that each side has a length of 1. If you draw a line from each of the 12 vertices to the centre, you sub-divide your dodecagon into twelve congruent isosceles triangles, each having a base of 1. Its area, of course is one-twelfth of the area of the dodecagon. Now, look at one of these twelve triangles. It has angles of 30°, 75°, and 75°. Consider its altitude as being the line perpendicular to the base of 1, extending from the midpoint of that base to the opposite vertex.

Now, let’s cut this triangle, along its altitude, into two smaller right triangles. Each of these smaller right triangles, then, has angles of 15°, 75°, and 90°. Its shortest side is equal to ½, in length. Its area is 1/24 of the area of the original dodecagon. The longer side, not the hypotenuse, has a length of ½ tan 75°. Thus, the area of this triangle may also be reckoned as ½ X the product of those two sides, namely:

½ X ½ X ½ X tan 75° = ⅛ tan 75°.

Thus, if A is the area of the dodecagon, then we have

1/24 A = ⅛ tan 75°; whence,

A = 3 tan 75°.

You can probably calculate in your head, tan 75° = 2 + √3; if not, then let me give you a hint:

By the sum-of-angles formula, we are given that

tan(A + B) = (tan A + tan B) / (1 – tan A tan B); that is,

tan 75° = (tan 45° + tan 30°) / (1 – tan 45° tan 30°)

= (1 + tan 30°) / (1 –tan 30°), since we know that tan 45° = 1.

On the other hand,

tan 30° = sin 30° / cos 30° = [ ½ ] / [ ½ √3 ] = ⅓√3.

Thus, we have

tan 75° = [ 1 + ⅓√3 ] / [ 1 –  Ã¢Â…“√3 ]

= (√3 + 1 ) / (√3 –  1 )  

= (√3 + 1 )² / [(√3 – 1 ) (√3 + 1) ]  

= ( 3 + 2√3 + 1² ) / ( 3 – 1²)

= ( 4 + 2√3 ) / 2

= ( 2 + √3 )

= 3.732, approx.

Thus, for a regular dodecagon whose side is 1, we have

A = 3 ( 2 + √3 );

and, for a regular dodecagon whose side is s, we have

A = 3 ( 2 + √3 ) s².

I hope that this helps you.