What is a polygon?

The word comes from Greek: poly means "many" and gōnia means "angle", so a polygon is a "many-angled" shape. Every polygon has the same number of sides, corners and angles: a pentagon has 5 of each.

Polygon definition: the four rules

A shape is a polygon if it passes all four of these tests.

1. It is flat. It lies in a plane, like a drawing on paper. A cube is a solid, not a polygon, though its faces are polygons.
2. Its sides are straight. Each side is a line segment. One curved edge and it isn't a polygon.
3. It is closed. The sides join up, end to end, with no gaps.
4. Its sides only meet at the ends. Two sides meet at each vertex, and the sides don't cross each other anywhere else.

Britannica puts it as a closed shape made of line segments connected so that no two cross. Some mathematicians also count shapes whose sides do cross, like a five-pointed star drawn in one line, as "complex" polygons. School maths sticks to the four rules above.

Polygons and non-polygons

Polygon names

Most polygon names use a Greek number followed by -gon. The table also gives each shape's interior angles, worked out with the formulas further down.

An 11-sided shape is a hendecagon, sometimes called an undecagon. Past 12 sides, people usually just say "a 15-gon" or "a 100-gon". A 100-gon is a polygon too, and each of its angles, if regular, is 176.4°: so close to a straight line that it looks like a circle.

Types of polygons

Regular and irregular polygons

A regular polygon has all its sides the same length and all its angles the same size. An equilateral triangle and a square are regular. An irregular polygon is any polygon that isn't. It needs both: a rhombus has four equal sides but its angles differ, so it is irregular, and a rectangle has four equal angles but its sides differ, so it is irregular too.

Convex and concave polygons

In a convex polygon, every interior angle is less than 180°, so all the corners point outwards. A concave polygon has at least one interior angle greater than 180°, a corner that points inwards, like the notch in an arrowhead. A quick test: if you can draw a straight line between two points inside the shape that passes outside it, the shape is concave. Every regular polygon is convex, and every triangle is convex, because a triangle's angles add up to only 180°.

Simple and complex polygons

A simple polygon has sides that only meet at the corners. A complex (or self-intersecting) polygon has sides that cross, like a pentagram drawn in five strokes. Unless a question says otherwise, "polygon" means a simple polygon.

Special triangles and quadrilaterals

• Triangles by sides: equilateral (3 equal), isosceles (2 equal) and scalene (none equal). By angles: acute, right-angled and obtuse.
• Quadrilaterals: square, rectangle, rhombus, parallelogram, trapezium (trapezoid in the US) and kite.

Interior angles of a polygon

Pick one corner and draw lines from it to every other corner it isn't already joined to. A polygon with n sides splits into n − 2 triangles, and each triangle's angles add up to 180°. So:

A pentagon splits into 3 triangles, which gives it an angle sum of 540°. This works for every simple polygon, regular or not, convex or concave. In a regular polygon all the angles are equal, so divide by n to get each one:

Exterior angles of a polygon

An exterior angle is the angle between one side and the next side extended outwards. At each corner the interior and exterior angles together make a straight line, 180°. Walk all the way round a convex polygon and you turn through one full turn, so:

The exterior angle is often the quickest route. For a regular hexagon, 360° ÷ 6 = 60°, and taking that from 180° leaves 120° for every interior angle.

Worked examples

Example 1: interior angles of a decagon

A decagon has 10 sides, so its interior angles add up to (10 − 2) × 180° = 8 × 180° = 1,440°.

Example 2: each angle of a regular octagon

The angles add up to (8 − 2) × 180° = 1,080°, and there are 8 equal angles, so each is 1,080° ÷ 8 = 135°. Check with the exterior angle: 360° ÷ 8 = 45°, and 180° − 45° = 135°.

Example 3: a missing angle in an irregular pentagon

A pentagon has angles of 100°, 120°, 95°, 110° and x. Find x.

A pentagon's angles add up to 540°. The four known angles add up to 425°, so x = 540° − 425° = 115°.

Example 4: how many sides?

A regular polygon has interior angles of 156°. How many sides does it have?

Use the exterior angle: 180° − 156° = 24°. The exterior angles add up to 360°, so n = 360 ÷ 24 = 15 sides. Can a regular polygon have interior angles of 100°? Its exterior angles would be 80°, and 360 ÷ 80 = 4.5. You can't have half a side, so no.

Example 5: area of a regular hexagon

Find the area of a regular hexagon with sides of 6 cm. Lines from the centre to each corner split it into 6 equilateral triangles with sides of 6 cm. One such triangle has area √3 ÷ 4 × 6² = 15.588 cm², so the hexagon's area is 6 × 15.588 = 93.53 cm² to 2 decimal places.

Splitting into triangles works for the area of any polygon. For a regular polygon there is also a formula: area = ½ × perimeter × apothem, where the apothem is the distance from the centre to the middle of a side.

How many diagonals does a polygon have?

A diagonal joins two corners that aren't next to each other. From each of the n corners you can draw a diagonal to every corner except itself and its two neighbours, which is n − 3 corners. That counts each diagonal twice, once from each end, so halve it:

A hexagon has 6 × 3 ÷ 2 = 9 diagonals, and a triangle has 3 × 0 ÷ 2 = 0, which is why a triangle can't be split into smaller triangles by joining corners.

Polygons in real life

• Stop signs in the US are regular octagons, and the federal Manual on Uniform Traffic Control Devices keeps the octagon for stop signs only, so the shape alone tells drivers what the sign means, even from behind.
• Honeycombs are made of hexagonal cells. Regular hexagons fit together with no gaps, as do equilateral triangles and squares, but no other regular polygon does.
• Floor tiles, footballs and building plans are full of polygons. A classic football is stitched from 12 pentagons and 20 hexagons.
• The UK 20p and 50p coins look like heptagons, but their sides are slightly curved, so they aren't polygons. The Royal Mint calls the shape an equilateral curve heptagon: the curves give it the same width in every direction, so it rolls smoothly in vending machines.

Polygons in computer graphics

Every 3D model in a video game or animated film is built from polygons, almost always triangles. A character can be made of tens of thousands of them, and the graphics card fills in each triangle's pixels many times a second. Triangles are used because three points always lie on one flat plane, so a triangle can never bend. Our Graphics Processing course shows how meshes of triangles become the images on your screen. Try counting the triangles in a mesh, then press Mark to check in the course.

Polygons at school

In the US, the Common Core standards ask grade 6 students to find the area of polygons by splitting them into rectangles and triangles, and polygon angles return in high school geometry. In England, the GCSE Maths content includes using the angle sum of a triangle to find the angle sum of any polygon and the properties of regular polygons. Daily papers for Geometry, Algebra 2 and SAT Math and GCSE Maths Higher give short timed practice with angle questions like the ones above.

Common mistakes

• Calling a circle a polygon. It has no straight sides, so it can't be one.
• Using n × 180° for the angle sum. It is (n − 2) × 180°, because the shape splits into n − 2 triangles, not n.
• Dividing by n for an irregular polygon. (n − 2) × 180° ÷ n only gives each angle when all the angles are equal.
• Mixing up interior and exterior angles. Interior is inside the corner. Exterior is the turn from one side to the next. They add up to 180°.
• Calling a rhombus regular. Regular needs equal sides and equal angles. Only the square manages both among quadrilaterals.

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