"How many sides does a pentagon have" is asked by a child in grade 2 learning shape names, by a parent checking a homework answer, and by anyone who has just read that a building is pentagon-shaped. The answer is five, and the interesting part is what follows from five: the angle sum, the number of diagonals, the symmetry, and the pattern that gives every polygon its name. This page has the pentagon's facts, the formulas that work for any polygon, the names chart from three sides to twenty and beyond, and where pentagons turn up in real life. The printable has the chart and practice questions on two pages.
Download the polygon names chart and pentagon facts sheet (PDF) (the pentagon's facts, the polygon chart from 3 to 20 sides with angle sums, regular angles and diagonals, the four formulas, the real-life examples, and twelve practice questions with answers; two pages, free).
The pentagon
A pentagon has five sides, five corners (vertices) and five interior angles. The name is Greek: penta, five, and gonia, angle.
| Fact | Any pentagon | Regular pentagon (all sides and angles equal) |
|---|---|---|
| Sides | 5 | 5, all the same length |
| Vertices (corners) | 5 | 5 |
| Interior angles | 5, adding to 540° | 108° each |
| Exterior angles | add to 360° (true of every polygon) | 72° each |
| Diagonals | 5 | 5, and they form a five-pointed star |
| Lines of symmetry | 0 to 5 | 5 |
| Rotational symmetry | depends | order 5 (it fits onto itself 5 times in a full turn) |
| Area | split into triangles | about 1.72 × side², exactly ¼√(5(5 + 2√5)) × side² |
| Tiles the floor alone? | some irregular ones do | no: 108° does not divide 360° |
Why 540°: draw lines from one corner to the two corners it does not touch, and the pentagon splits into three triangles. Each triangle's angles add to 180°, so the pentagon's add to 3 × 180° = 540°. This is the method for every polygon: a shape with n sides splits into n − 2 triangles.
The 108° and the 72°: in a regular pentagon the 540° is shared equally, 540° ÷ 5 = 108°. The exterior angle at each corner (the turn you make walking round it) is 180° − 108° = 72°, and five turns of 72° make the full 360°.
Kinds of pentagon:
- Regular: five equal sides, five angles of 108°. The one in textbooks.
- Irregular: five sides of any lengths; the angles still add to 540°. Home plate in baseball is one: a 17-inch front edge, two short sides at right angles to it, and two longer sides meeting at a right angle at the back point, so its angles are 90°, 90°, 135°, 135° and 90°.
- Convex: every angle is less than 180° and every diagonal stays inside. Concave: at least one angle is bigger than 180°, so the shape has a dent; a chevron-shaped arrow with five sides is a concave pentagon.
Where pentagons appear
- Home plate in baseball: a 17-inch square with two corners cut off, five sides.
- The Pentagon, the headquarters of the US Department of Defense outside Washington, is a regular pentagon with each side about 921 feet long.
- A football (soccer ball): the classic ball has 12 black pentagons and 20 white hexagons (a truncated icosahedron).
- Okra cut across, a starfish's outline, many flowers with five petals, and the five-pointed star made by a regular pentagon's diagonals.
- Road signs: the US school-zone sign is a pentagon with its point upwards.
The formulas for any polygon
With n the number of sides:
| What | Formula | For a pentagon |
|---|---|---|
| Sum of interior angles | (n − 2) × 180° | 3 × 180° = 540° |
| Each interior angle, regular | (n − 2) × 180° ÷ n | 108° |
| Each exterior angle, regular | 360° ÷ n | 72° |
| Number of diagonals | n(n − 3) ÷ 2 | 5 × 2 ÷ 2 = 5 |
| Number of triangles from one corner | n − 2 | 3 |
The exterior angles of any convex polygon add to 360°, however many sides it has, because walking round it turns you through one full circle.
The polygon names chart
| Sides | Name | Angle sum | Each angle (regular) | Diagonals |
|---|---|---|---|---|
| 3 | triangle | 180° | 60° | 0 |
| 4 | quadrilateral (square, rectangle, rhombus, trapezium, kite, parallelogram) | 360° | 90° | 2 |
| 5 | pentagon | 540° | 108° | 5 |
| 6 | hexagon | 720° | 120° | 9 |
| 7 | heptagon (septagon) | 900° | 128.6° | 14 |
| 8 | octagon | 1,080° | 135° | 20 |
| 9 | nonagon (enneagon) | 1,260° | 140° | 27 |
| 10 | decagon | 1,440° | 144° | 35 |
| 11 | hendecagon (undecagon) | 1,620° | 147.3° | 44 |
| 12 | dodecagon | 1,800° | 150° | 54 |
| 13 | tridecagon | 1,980° | 152.3° | 65 |
| 14 | tetradecagon | 2,160° | 154.3° | 77 |
| 15 | pentadecagon | 2,340° | 156° | 90 |
| 16 | hexadecagon | 2,520° | 157.5° | 104 |
| 17 | heptadecagon | 2,700° | 158.8° | 119 |
| 18 | octadecagon | 2,880° | 160° | 135 |
| 19 | enneadecagon | 3,060° | 161.1° | 152 |
| 20 | icosagon | 3,240° | 162° | 170 |
| 30 | triacontagon | 5,040° | 168° | 405 |
| 100 | hectogon | 17,640° | 176.4° | 4,850 |
| 1,000 | chiliagon | 179,640° | 179.64° | 498,500 |
| n | n-gon | (n − 2) × 180° | (n − 2) × 180° ÷ n | n(n − 3) ÷ 2 |
Above twelve sides, mathematicians usually say "13-gon" or "20-gon" rather than the Greek names. The regular polygons with 3, 4 and 6 sides are the only ones that tile a flat floor by themselves (their angles, 60°, 90° and 120°, divide 360°); a regular pentagon's 108° does not, which is why bathroom tiles are never regular pentagons and why a football needs hexagons between its pentagons.
Where it sits in school
Naming shapes by their sides is kindergarten to grade 2 in the United States (Common Core K.G and 2.G.1 ask for pentagons and hexagons by name), England's Year 1 and 2, and CBSE Classes 1 to 3. Angle sums and the (n − 2) × 180° rule are grade 7 and 8 (and GCSE and Class 8 Understanding Quadrilaterals); the diagonal formula and regular-polygon area are high-school geometry. The exam error at every level is the same: calling any five-sided shape "not a pentagon" because it is not the regular one.
Practice: twelve questions
- How many sides, corners and angles does a pentagon have?
- What do the interior angles of a pentagon add to?
- Each angle of a regular pentagon is how big?
- Four angles of a pentagon are 100°, 110°, 120° and 90°. What is the fifth?
- How many diagonals does a pentagon have?
- What is the exterior angle of a regular pentagon?
- Name the polygons with 6, 7, 8 and 10 sides.
- What is the angle sum of a hexagon?
- How many diagonals does an octagon have?
- Each angle of a regular polygon is 150°. How many sides does it have?
- Is a shape with five sides, one of which is curved, a pentagon?
- A regular pentagon has sides of 10 cm. Estimate its area.
Answers: 1) 5, 5, 5. 2) 540°. 3) 108°. 4) 540 − 420 = 120°. 5) 5. 6) 72°. 7) Hexagon, heptagon, octagon, decagon. 8) 720°. 9) 8 × 5 ÷ 2 = 20. 10) Exterior angle 30°, so 360 ÷ 30 = 12 sides, a dodecagon. 11) No: a polygon's sides must all be straight. 12) About 1.72 × 100 = 172 cm².
Download the polygon names chart and pentagon facts sheet (PDF)
Where to get help
A 1-on-1 online maths class with a tutor who holds a master's in mathematics, for grades 2 to 8 in the US curriculum and the equivalent years in England, Australia and CBSE: shapes built and named on the shared screen for the youngest, the angle-sum rule derived by cutting a pentagon into triangles for the older ones, then the exam questions. The recording and notes after every class, the same tutor every week, US$10 to $13 an hour. The grade 6 math page has the grades; the free 30-minute first class is a real lesson, usually on the homework the child brings.
Questions parents ask
1How many sides does a pentagon have?
2How many angles does a pentagon have?
3How many diagonals does a pentagon have?
4What is the difference between a regular and an irregular pentagon?
5How many sides does a hexagon have, and a heptagon, and an octagon?
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