Answer:
A = 742.3 square units
Step-by-step explanation:
Draw the figure you have given. Draw a perpendicular from the given center point to the horizontal side above. Now you have a right triangle. The hypotenuse is 16.2.
The sum of the measures of the angles of a an n-sided polygon is
180(n - 2)
This polygon has 8 sides, so n = 8
180(n - 2) = 180(8 - 2) = 180(6) = 1080
The sum of the measures of the interior angles is 1080 deg.
There are 8 sides and 8 interior angles. Since this is a regular octagon, all sides are congruent, and all interior angles are congruent.
Each interior angle measures 1080/8 = 135 deg.
The angle on the upper right side of the right triangle is 135/2 = 67.5 deg.
For that angle, the segment you drew is the opposite leg.
The trig ratio that relates the opposite leg and the hypotenuse is the sine.
[tex] \sin A = \dfrac{opp}{hyp} [/tex]
[tex] \sin 67.5^\circ = \dfrac{x}{16.2} [/tex]
[tex] x = 16.2 \sin 67.5^\circ [/tex]
[tex] x = 14.9668 [/tex]
Now we find the length of half of one side of the octagon.
a^2 + b^2 = c^2
14.9668^2 + b^2 = 16.2^2
c = 6.2
Half of a side of the octagon measures 6.2; the entire side of the octagon measures 12.4.
An octagon is made up of 8 congruent triangles.
A = 8 * bh/2
A = 8 * 12.4 * 14.97/2
A = 742.3 square units
