Octagon Calculator
Calculate area, perimeter, interior angle, and apothem of a regular octagon from its side length.
How it works
- 1Enter the side length of your regular octagon.
- 2The area is computed as 2(1 + √2) · side², derived from the general formula ½ · perimeter · apothem.
- 3Interior angle = (8 − 2) × 180° ÷ 8 = 135°; apothem = side ÷ (2 · tan(π/8)).
Use cases
- Architectural and carpentry projects involving octagonal floor tiles, windows, or frames.
- Engineering calculations for stop-sign shapes or octagonal bolt heads.
- Geometry coursework verifying hand-calculated polygon properties.
Frequently asked questions
Why is the interior angle of a regular octagon always 135°?
The sum of interior angles of any n-gon is (n − 2) × 180°. For n = 8 that is 1080°, and dividing by 8 gives 135° per angle.
What is the apothem?
The apothem is the perpendicular distance from the centre of the octagon to the midpoint of any side — equal to side ÷ (2 · tan(π/8)).
How is the area formula 2(1 + √2) · side² derived?
Substituting n = 8 into ½ · n · side · apothem and simplifying tan(π/8) = √2 − 1 yields the compact form 2(1 + √2) · side².
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