Octagon Calculator
Find a regular octagon's area, perimeter, width across flats and corners, and apothem from its side length.
Area
120.7107
Area = 2(1 + √2) × s². A side of 5 covers about 120.71 square units.
Perimeter (8s)
40
Width across flats
12.0711
Width across corners
13.0656
Apothem
6.0355
How it works
A regular octagon has eight equal sides and eight equal angles of 135° each. One measurement — the side length — is enough to pin down its area, its perimeter, and how wide it sits in either direction.
The area uses the tidy formula 2(1 + √2) × s², and the perimeter is eight times the side. Two widths matter in practice: the distance across the flat sides is (1 + √2) × s, while the distance across opposite corners is √(4 + 2√2) × s. The apothem, from the center to a side, is half the across-flats width.
Stop signs, umbrella frames, and gazebo floors are everyday octagons. Enter a side of 5 and the area works out to about 120.71 square units.
Frequently asked questions
What's the area formula for a regular octagon?
Area = 2(1 + √2) × s², where s is the side length. The 2(1 + √2) factor is roughly 4.83, so an octagon's area is close to five times the square of its side.
What's the difference between width across flats and across corners?
Across the flats is the distance between two opposite sides, (1 + √2) × s, which is what you'd measure with calipers on a bolt head. Across the corners is the longer distance between two opposite vertices, √(4 + 2√2) × s.
How big is each interior angle?
Every interior angle of a regular octagon is 135°. The eight angles add up to 1080°, which is (8 − 2) × 180° by the polygon angle-sum rule.