Regular Polygon Calculator
Pick the number of sides and enter the side length to get the area, perimeter, angles, apothem, circumradius and diagonals, with a drawing. Advanced solves from any one measurement and adds frame cuts, cost and irregular shapes.
Any whole number from 3 to 1,000.
The length of one edge. Fractions like 3 1/2 work too.
Step-by-step working
How your polygon is worked out
Cut the shape into equal triangles from the centre. Each one is a side wide and an apothem tall.
- 1
Cut into triangles
360° ÷ 6
= 60° each - 2
Find the apothem
2 ÷ (2 × tan 30°)
a = 1.73205 cm - 3
Add up the area
½ × 12 × 1.73205
A = 10.3923 cm² - 4
Corner angles
(6 − 2) × 180 ÷ 6
120° inside
Read more: the formulas, a worked example, frames and the shoelace formula
A regular polygon has all its sides the same length and all its corners the same angle. Lines from the centre to each corner cut it into n equal triangles. Each triangle is one side wide, and its height is the apothem (centre to the middle of a side). The circumradius is the distance from the centre to a corner. Once you know n and any one length, everything else follows.
The formula
area A = n × s² ÷ (4 × tan(180° ÷ n)) = ½ × perimeter × apothem
perimeter P = n × s
apothem a = s ÷ (2 × tan(180° ÷ n))
circumradius R = s ÷ (2 × sin(180° ÷ n))
interior angle = (n − 2) × 180° ÷ n
exterior angle = central angle = 360° ÷ n
sum of interior angles = (n − 2) × 180°
diagonals = n × (n − 3) ÷ 2
side from apothem: s = 2 × a × tan(180° ÷ n)
side from circumradius: s = 2 × R × sin(180° ÷ n)
side from area: s = √(4 × A × tan(180° ÷ n) ÷ n)
frame miter angle = 180° ÷ n; inside edge = s − 2 × board width × tan(180° ÷ n)
shoelace area = ½ × |(x₁y₂ − x₂y₁) + (x₂y₃ − x₃y₂) + … + (xₙy₁ − x₁yₙ)|
Worked example
A hexagon (n = 6) with sides of 2 cm. 180° ÷ 6 = 30°, tan 30° = 0.577350 and sin 30° = 0.5.
- Perimeter = 6 × 2 = 12 cm.
- Apothem = 2 ÷ (2 × 0.577350) = 2 ÷ 1.154701 = 1.73205 cm (exactly √3).
- Circumradius = 2 ÷ (2 × 0.5) = 2 ÷ 1 = 2 cm. On a hexagon the corners are one side away from the centre.
- Area = 6 × 2² ÷ (4 × 0.577350) = 24 ÷ 2.309401 = 10.3923 cm² (exactly 6√3). Check: ½ × 12 × 1.73205 = 10.3923.
- Interior angle = (6 − 2) × 180 ÷ 6 = 720 ÷ 6 = 120°; exterior angle = 360 ÷ 6 = 60°; the six corners add up to 720°.
- Diagonals = 6 × (6 − 3) ÷ 2 = 18 ÷ 2 = 9.
The default irregular polygon in Advanced is a house shape with corners (0, 0), (6, 0), (6, 4), (3, 7), (0, 4) in metres. The shoelace terms are 0 × 0 − 6 × 0 = 0, 6 × 4 − 6 × 0 = 24, 6 × 7 − 3 × 4 = 30, 3 × 4 − 0 × 7 = 12 and 0 × 0 − 0 × 4 = 0. They add up to 66, so the area is 66 ÷ 2 = 33 m²: a 6 × 4 rectangle (24 m²) plus a roof triangle of 6 × 3 ÷ 2 = 9 m².
Frames, staves and nuts
For a frame with n pieces, set the miter saw to 180° ÷ n on both ends of every piece: 60° for a triangle, 45° for a square, 36° for a pentagon, 30° for a hexagon and 22.5° for an octagon. For a hexagon frame with a 30 cm outside edge and 4 cm wide boards, each piece is 30 cm on its long (outside) edge and 30 − 2 × 4 × 0.577350 = 30 − 4.61880 = 25.3812 cm on its short (inside) edge, and you need 6 × 30 = 180 cm of board plus the blade allowance. Staves for a bucket or barrel use the same angle as a bevel. Spanners and nuts are sized across the flats (2 × apothem); across the corners is 2 × circumradius.
Material cost
A hexagon patio with 2 m sides covers 10.3923 m². With 10% extra for cuts that is 10.3923 × 1.10 = 11.4315 m², and at 25 a square metre it costs 11.4315 × 25 = 285.79 in your currency.
Polygon names
3 triangle, 4 square, 5 pentagon, 6 hexagon, 7 heptagon, 8 octagon, 9 nonagon, 10 decagon, 11 hendecagon, 12 dodecagon, 20 icosagon, 100 hectogon, 1,000 chiliagon. Other sizes are called n-gons, such as a 15-gon. As n grows, the polygon gets closer to a circle: a 1,000-gon with circumradius 1 has an area of 3.14157, against π = 3.14159.
Standard geometry, no rates or rules that change. For circles and arcs use the circle calculator; for triangles that are not equilateral, the triangle calculator; for other shapes, the area calculator.
Common questions
How do I find the area of a regular polygon?
Multiply the number of sides by the side squared, then divide by 4 × tan(180° ÷ n): A = n × s² ÷ (4 × tan(180°/n)). A hexagon with 2 cm sides gives 6 × 4 ÷ (4 × tan 30°) = 24 ÷ 2.3094 = 10.3923 cm², which is exactly 6√3. Another way is half the perimeter times the apothem: ½ × 12 × 1.73205 = 10.3923.
How do I work out the interior angle of a polygon?
Take 2 away from the number of sides, multiply by 180 and divide by the number of sides: (n − 2) × 180 ÷ n. An octagon has (8 − 2) × 180 ÷ 8 = 135° at each corner, and a hexagon has 120°. All the interior angles add up to (n − 2) × 180, so 1,080° for an octagon.
What is the apothem?
The apothem is the distance from the centre straight out to the middle of a side, at right angles to it. It is also the radius of the biggest circle that fits inside (the inradius). For side s it is s ÷ (2 × tan(180°/n)). On a hexagon nut it is half the distance across the flats.
What angle do I cut for a hexagon or octagon frame?
Set the miter saw to 180 ÷ n on both ends of every piece: 30° for a hexagon, 22.5° for an octagon, 45° for a square and 36° for a pentagon. Each joint then closes at the polygon's interior angle. Advanced also gives the short-point length on the inside edge for your board width.
How many diagonals does a polygon have?
n × (n − 3) ÷ 2. Each corner joins to every other corner except itself and its two neighbours, and each diagonal gets counted twice. A hexagon has 6 × 3 ÷ 2 = 9 diagonals; a triangle has none.
How do I find the area of an irregular polygon?
List the corners in order as x, y coordinates and use the shoelace formula: multiply each x by the next y, take away the next x times this y, add these up and halve the total. Switch to Advanced and type the corners into the irregular polygon box; it also gives the perimeter and the centre point.
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