A square is formed by joining the midpoints of alternate sides of a regular octagon with side length 14 in. Find the shaded area.

98(1 + 2√2) in² ≈ 375 in²

Step-by-step explanation:

Assuming the shaded region is outside of the square and inside of the octagon, we can find the area by subtracting the area of the square from the area of the octagon.

The area of a regular octagon is 2 (1 + √2) s².  We can show this by finding the area of the square outside of the octagon, and subtracting the triangles in the corners:

(s + √2 s)² − 4 (½ (½√2 s)²)

(1 + √2)² s² − 4 (½ (s²/2))

(1 + 2√2 + 2) s² − s²

2 (1 +√2) s²

The diagonal of the inner square is equal to the width of the octagon, (1+√2) s.  So the side length of the square is:

½√2 (1+√2) s

½(2+√2) s

The area of the square is therefore:

(½(2+√2) s)²

¼(2+√2)² s²

¼(4+4√2+2) s²

¼(6+4√2) s²

½(3+2√2) s²

The area of the shaded region is therefore:

2 (1 +√2) s² − ½(3+2√2) s²

½ s² (4 (1 +√2) − (3+2√2))

½ s² (4 + 4√2 − 3 − 2√2)

½ s² (1 + 2√2)

The side length of the octagon is s = 14 in, so the area is:

½ (14 in)² (1 + 2√2)

= 98(1 + 2√2) in²

≈ 375 in²