Math
Octagon Calculator
The octagon calculator solves a regular octagon — the stop-sign shape — from one known measurement.
Its area is exactly 2(1 + √2)·s² ≈ 4.8284·s², its apothem is (1 + √2)s/2, and every interior angle is 135°; the calculator reports all of them along with the perimeter and circumradius.
Apothem = center to the middle of a side. Circumradius = center to a vertex.
Area is in square units; the rest are lengths.
Pick the unit your measurements use. Area is reported in square units and volume in cubic units.
Try an example
Result
Area
—
- Side
- —
- Perimeter
- —
- Apothem (inradius)
- —
- Circumradius
- —
- Interior angle
- —
- Exterior angle
- —
More details (1 more)
- Number of diagonals
- —
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Study path
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What the octagon calculator solves
The octagon calculator solves a regular octagon — the stop-sign shape — from one known measurement. Its area is exactly 2(1 + √2)·s² ≈ 4.8284·s², its apothem is (1 + √2)s/2, and every interior angle is 135°; the calculator reports all of them along with the perimeter and circumradius.
Formula
How to use it
- Choose which octagon measurement you know: side, apothem, circumradius, area, or perimeter.
- Enter the value and choose its unit.
- Read the area, side, perimeter, apothem, circumradius, angles, and diagonal count; open the work for each formula.
How to read the answer
A regular octagon has 8 equal sides and 8 equal angles of 135°. The apothem is the radius of the circle that fits inside it and the circumradius is the radius of the circle through its corners; the area is half the perimeter times the apothem.
Common mistakes and edge cases
- Mixing up apothem (center to the middle of a side) with circumradius (center to a corner). The circumradius is always longer.
- Using the interior angle formula with the wrong n. Interior angle = (n − 2) × 180° / n.
- Forgetting that area is in square units while every other result is a length.
- Zero or negative measurements cannot form a polygon and return an error.
Worked examples
Side 3
Area
43.4558
Apothem 6 cm
Area
119.2935
Circumradius 10 in
Area
282.8427
Area 200 square units
Area
200
Perimeter 40 ft (side 5)
Area
120.7107
Decimal side 2.5 m
Area
30.1777
Zero apothem (impossible)
Area
Error
Frequently asked questions
What is the area of a regular octagon?+
A = 2(1 + √2)·s² ≈ 4.828·s². Side 3 gives about 43.46. You can also use ½ × perimeter × apothem.
What are the angles of an octagon?+
Each interior angle of a regular octagon is 135° (total 1080°) and each exterior angle is 45°.
How many diagonals does an octagon have?+
Twenty: n(n − 3)/2 = 8 × 5 / 2.
How wide is an octagon across the flats?+
Twice the apothem: (1 + √2)·s ≈ 2.414·s. Across the corners it is twice the circumradius, about 2.613·s. A stop sign with 30 in sides is about 72.4 in across the flats.
How do I find the side of an octagon from its area?+
s = sqrt(A / (2(1 + √2))) = sqrt(A / 4.8284). Area 200 gives side ≈ 6.436.
How do I cut an octagon from a square?+
Cut off each corner of a square of side L at 45°; the piece to remove has legs L / (2 + √2) ≈ 0.2929·L. The octagon's side is then L·(√2 − 1) ≈ 0.4142·L.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026