What's is the area of a regular octagon with a radius of 12?

Area = 672.62sq2

Step-by-step explanation:

The sum of all octagon angles is 1080 degrees, which makes each angle equal to 135 degrees in measure. However we can cut into 4 which is easier and use one triangle angle as 2 triangles

We use trigonometry and find a right angle in the triangle cuts and know finding area of one triangle we just multiply by 8 after to find the area.

L 1 = 12

Angle = 135/2 = 67.5 degree to make the 2nd triangle = right angle triangle. like quarter of the octagon 8.2 = 4 = 1/4 of the Octagon = 360/4 = 90 angle etc.

Sin (67.5) = 0.9238795325

12 / 0.9238795325 = 12.9887064037 We know have 2 sides and need the third side that make up the two combined triangles.

12sq = 144

12.9887064037 x 12.9887064037 = 168.706494042sqrt

144 + 12.9887064037 = 56.9887064037

2 triangle abc/abc = 12.9887064037, 56.9887064037, and 12

We know 56.99 we know this represents 2 sides but have to half and get back to one triangle, or the area becomes off sync with the way the hexagon angles jut out.

So 56.9887064037/2 = 28.4943532019/2

We know cna find height using 28.49/2 base here by dividing by 2. then using half a normal angle of hexagon = 135 degree

= 67.5 into the calculator.

14.247176601 / tan (67.5) (O/H) = 5.901373774

A = H x (Half Base) = 14.247176601 x 5.901373774 = 84.0779143467 x 8

then multiply by 8 = 672.623314774

Area = 672.62sq2