The general formula for the area of a regular n-gon is:
A = 0.5•ap
Where:
A = area of the regular n-gon.
a = apothem = it is the distance from the center of n-gon to the midpoint of one side.
p = perimeter of n-gon = the sum of all of the measure of its sides.
*The apothem and the side form a 90° angle.
The measure of one interior angle of a regular n-gon = (n-2)•180° / n
OC = apothem
BD = one side of 24-gon
OB = OD = radius
m∠OBD = m∠ODB = half of the measure of one interior angle
m∠OCD =m∠OCB = 90°
m∠OBD = m∠ODB = 0.5(24-2)(180°) / 24 = 82.5°
sin (∠OBD) = a / OB
Note: the meaning of sine of an angle is opposite side of the angle over the hypotenuse of a right triangle.
sin (82.5°) = a /14
multiply 14 on both sides:
14•sin (82.5°) = a
a ≈ 13.88 mm
cos (82.5º) = BC/14
BC = 14•cos (82.5º)
BC ≈ 1.83 mm
BD = 2•BC ≈ 2 (1.83) ≈ 3.65 mm∴
p = 3.65(24) = 87.71 mm
Now let's compute the area of 24-gon.
A = 0.5 (13.88)(87.71) ≈ 608.74 mm2
Note: In your calculator, make sure you round off only the final answer to make it more accurate.
