Regular polygons
A regular polygon has all its sides the same length and all its interior angles the same size. An equilateral triangle, a square and the hexagon of a honeycomb cell are regular polygons.
The interior angles of any polygon with n sides add up to (n − 2) × 180°, because the polygon can be cut from one corner into n − 2 triangles. A pentagon makes 3 triangles, so its angles add up to 3 × 180 = 540°.
A pentagon cut from one corner makes three triangles, so its interior angles add up to 3 × 180° = 540°.
Share the total equally
In a regular pentagon the five interior angles are all the same, so they share the 540° equally. Each one is 540 ÷ 5 = 108°. Check: 5 × 108 = 540.
A hexagon cut from one corner makes 4 triangles, so its angles add up to 4 × 180 = 720°, and each of its six angles is 720 ÷ 6 = 120°.
For any regular polygon with n sides, each interior angle is . An equilateral triangle has 180 ÷ 3 = 60° at each corner, a square 360 ÷ 4 = 90°, and a regular octagon 1080 ÷ 8 = 135°. This works only for a regular polygon: in any other polygon the angles can be different sizes, and only their total is fixed.
A regular pentagon: five equal interior angles of 108°, and 5 × 108° = 540°.
A regular hexagon: six equal interior angles of 120°, and 6 × 120° = 720°.
Exterior angles
Make one side of the polygon longer, past the corner where it ends. The angle between that extended side and the next side of the polygon is the exterior angle at that corner.
The interior angle and the exterior angle at a corner sit side by side on the straight extended side, so together they make 180°. At a corner of a regular pentagon, the exterior angle is 180 − 108 = 72°.
The exterior angle is also a turn. Walk along one side of the polygon. At the corner, to set off along the next side, you have to turn, and the angle you turn through is the exterior angle.
Each side of the regular pentagon is carried on past its corner. The angle between the extended side and the next side is the exterior angle.
One full turn
Walk all the way round the polygon, turning at each corner, until you are back where you started and facing the way you first set off. Every turn was the same way, and together they have taken you exactly once round: 360°. The turns are the exterior angles, so the exterior angles add up to 360°. This is true of every polygon without a dent, whatever its number of sides.
A regular pentagon has five equal exterior angles, so each is 360 ÷ 5 = 72°, the same 72° found from 180 − 108.
The five exterior angles of a regular pentagon are 72° each, and 5 × 72° = 360°.
walking round any polygon turns you through one full turn, so the exterior angles sum to one full turn = 360° at every size
Shrink the polygon to a point
This pentagon is not regular, and its five exterior angles are marked. Drag the scale handle to shrink the pentagon toward a point without turning any side: the five angles do not change, and at a point they close up into one full circle, 360°.
The quicker method
The exterior angles give a shorter route to the interior angle of a regular polygon. First share 360° between the n corners: each exterior angle is . Then take that from 180°: each interior angle is .
A regular decagon has 10 sides. Each exterior angle is 360 ÷ 10 = 36°, so each interior angle is 180 − 36 = 144°. The long way agrees: (10 − 2) × 180 ÷ 10 = 1440 ÷ 10 = 144°.
The method also runs backward. If each interior angle of a regular polygon is 140°, each exterior angle is 180 − 140 = 40°, and the number of sides is 360 ÷ 40 = 9.
The usual mistakes
Giving the total for one angle. 540° is the sum of all five angles of a pentagon; one angle of a regular pentagon is 540 ÷ 5 = 108°.
Giving the exterior angle for the interior one. 360 ÷ 5 = 72° is the turn at each corner of a regular pentagon; the angle inside the corner is 180 − 72 = 108°.
Sharing 180° instead of 360°. The turns all the way round make a full turn, 360°, not a half turn: 180 ÷ 5 = 36° is not an angle of a regular pentagon.
Worked example: Exterior Angles Add to 360°
Question Each interior angle of a regular polygon is 156°. How many sides does it have? A different regular polygon has 20 sides; find each of its interior angles.
1.Exterior angle at each vertex = 180° − 156° = 24°.
At each corner, the exterior angle is 180° minus the interior one. 2.The exterior angles of any polygon add to 360°, and here they are all equal.
Walk round the polygon: the turns add to one full circle, 360°. 3.Number of sides = 360° ÷ 24° = 15.
Equal turns of 24°: 360° ÷ 24° = 15 sides. Slide to 15 to see it. 4.For 20 sides: exterior angle = 360° ÷ 20 = 18°.
Twenty sides: each turn is 360° ÷ 20 = 18°. 5.Interior angle = 180° − 18° = 162°.
Interior angle = 180° − 18° = 162°.
Answer: 15 sides; 162°
Common mistakes
- Dividing 360° by the interior angle: 360 ÷ 156 is not a whole number, which is the sign the wrong angle was used.
- Giving the 20-gon's exterior angle, 18°, as its interior angle.
Worked example: Fitting Polygons Round a Point
Question Two regular hexagons and some equilateral triangles are placed round a point with no gaps and no overlaps. How many triangles are there? Can regular pentagons alone be fitted round a point in the same way?
1.Each angle of a regular hexagon is 120°: two hexagons take 240°.
Two hexagon corners at the point: 120° + 120° = 240°. 2.What is left: 360° − 240° = 120°.
The gap: 360° − 240° = 120°. 3.Each angle of an equilateral triangle is 60°: 120° ÷ 60° = 2 triangles.
Two triangle corners of 60° fill it exactly. 4.Regular pentagons: 360° ÷ 108° is not a whole number. Three pentagons take 324° and leave a 36° gap; a fourth would overlap.
Three pentagon corners: 324°, and a 36° gap a fourth cannot fit. 5.So pentagons alone cannot fit round a point.
Pentagons alone never close round a point.
Answer: 2 triangles; no, three pentagons leave a 36° gap
Common mistakes
- Fitting the shapes by eye and believing a picture: only the angle sum decides.
- Using 180° at the point, as if the shapes sat on a straight line rather than all the way round.