Question

Octagonal houses were popular in the 19th century one reason was that an octagon with the same perimeter as a square encloses a greater area than the square. To the nearest square ft, find the areas of an octagon and a square with perimeters of 80 ft.

Answer 1

• octagon: 483 ft²
• square: 400 ft²

Explanation:

You want the areas of an octagon and a square, each with a perimeter of 80 ft.

Square

The side length of a square is 1/4 of its perimeter, so the square of interest has a side length of ...

(80 ft)/4 = 20 ft

The area is the square of the side length, so the area of the square is ...

A = s²

A = (20 ft)² = 400 ft²

Octagon

A regular octagon has 8 sides of equal length, so the side length is ...

(80 ft)/8 = 10 ft

The area is found by the formula ...

A = 2(1 +√2)s²

A = 2(1 +√2)(10 ft)² ≈ 483 ft²

The area of the octagon is about 483 square feet; about 400 square feet for the square.

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