Question

What is the area of a regular octagon with a side of $ 8 $ inches?

Answer 1

Hint: For solving the above question we should know about the basics of octagon and a regular figure. Octagon is a convex polygon which has $ 8 $ sides. In the above question we are given that it is a regular octagon, this means that all the sides of the octagon are equal and all the angles are equal. We have the length of the side of an octagon. The formula of area of a regular octagon is $ 2{a^2}(1 + \sqrt 2 ) $ .

Complete step by step solution:
Before solving the question, let us draw the image of a regular octagon:

As we can see in the above image it has eight sides which are all equal. One side of the octagon is given in the question which is $ 8 $ inches.
Now we know the formula for the area of a regular octagon is $ 2{a^2}(1 + \sqrt 2 ) $ .
Here $ a = 8 $ , the side of the octagon. By putting the value in the formula we have
$ 2 \times {8^2}(1 + \sqrt 2 ) $ .
We will now solve the value Area $ = 2 \times 64(1 + \sqrt 2 ) $ . We know the approx. the value of $ \sqrt 2 $ is $ 1.414 $ .
Putting the value in the formula we have,
Area $ = 128(1 + 1.414) = 128 \times 2.414 $ .
So it gives us the Area of the regular octagon $ 308.992 $ .
Hence the required area is $ 309.02(approx) $ inches.
So, the correct answer is “ $ 309.02(approx) $ square inches”.

Note: We should know that all the angles of a regular octagon is $ {135^ \circ } $ . All the internal angles are always less than $ 180 $ degrees. Before solving this kind of question we should be fully aware of the formula of a regular octagon and its properties. We can find the area of an octagon by dividing the figure into portions of polygons whose area is well known.