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Difficulty: MediumPythagorean Theorem and Special Right Triangles

A regular hexagon ABCDEFABCDEF has a perpendicular distance of 12312\sqrt{3} inches between its two parallel opposite sides. What is the perimeter, in inches, of hexagon ABCDEFABCDEF?

Answer: 72 inches

Answer

The perimeter of the regular hexagon is 72 inches.
A regular hexagon with side length ss can be partitioned from its center into 6 congruent equilateral triangles of side length ss. Dropping an altitude from the center to any side creates two 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangles. In each right triangle, the side opposite the 60∘60^\circ angle (the altitude) has length s32\frac{s\sqrt{3}}{2}. The total perpendicular distance between two parallel opposite sides of the hexagon equals twice this altitude, s3s\sqrt{3}. Setting s3=123s\sqrt{3} = 12\sqrt{3} gives s=12s = 12 inches. The perimeter of the regular hexagon is 6×12=726 \times 12 = 72 inches.

Step-by-Step Solution

1
Express the perpendicular distance between parallel opposite sides of a regular hexagon in terms of its side length ss.
The distance between opposite sides is s3s\sqrt{3}.
A regular hexagon with side length ss consists of 6 congruent equilateral triangles. The altitude of each equilateral triangle divides it into two 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangles with legs s/2s/2 and s32\frac{s\sqrt{3}}{2}. The distance between opposite parallel sides spans two altitudes, which equals 2×s32=s32 \times \frac{s\sqrt{3}}{2} = s\sqrt{3}.
2
Solve for the side length ss.
s=12s = 12 inches.
Equating the given distance 12312\sqrt{3} to s3s\sqrt{3} yields s=12s = 12.
3
Calculate the total perimeter of the hexagon.
Perimeter =72= 72 inches.
A regular hexagon has 6 equal sides, so its perimeter is 6×12=726 \times 12 = 72 inches.

Key Concept

Applying 30∘−60∘−90∘30^\circ-60^\circ-90^\circ special right triangle relationships to regular polygons
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Topics

Pythagorean Theorem and Special Right Triangles

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