Angle Sum of Polygon
A polygon with \(n\) sides can be split into \(n - 2\) triangles from one vertex, so its interior angles add to \((n - 2) \times 180^\circ\). In NSW Year 8 maths this subtopic takes the angle sums of triangles and quadrilaterals from Stage 4 and extends them to any polygon, which looks ahead to Stage 5.
You will name polygons from the pentagon to the dodecagon, find the angle sum of shapes such as a decagon or a 24-gon, and find one interior angle of a regular polygon. You will also use the fact that the exterior angles add to \(360^\circ\), work backwards from an angle to the number of sides, find missing angles in irregular polygons, and decide which regular tiles can cover a floor without gaps. It is part of Year 8 Geometry.
Theory
A polygon is a closed shape made of straight sides. The diagonals from one vertex split a polygon with \(n\) sides into \(n - 2\) triangles, so its angles add to \(S = (n - 2) \times 180^\circ\). A pentagon has \(S = 3 \times 180 = 540^\circ\). This Year 8 maths page shows how.
Interior angle sum. Draw every diagonal from one vertex. Each triangle gives \(180^\circ\), so \(S = (n - 2) \times 180^\circ\) (angle sum of a polygon).
Regular polygon: all sides equal and all angles equal. One interior angle is \(\theta = \dfrac{(n - 2) \times 180}{n}\) degrees.
Exterior angles. Extend each side the same way around. The exterior angles of a convex polygon add to \(360^\circ\), whatever \(n\) is. In a regular polygon each is \(360 \div n\), and at every corner interior \(+\) exterior \(= 180^\circ\).
Working backwards to \(n\). From the angle sum: divide by \(180\), then add \(2\). From one interior angle of a regular polygon: take it from \(180^\circ\) to get the exterior angle, then divide \(360^\circ\) by that.
A missing angle. Add the known angles and subtract from \(S\), collecting like terms when angles are marked the same. A reflex angle at a dent is one of the interior angles.
Tiling. Tiles cover a floor with no gaps only if the angles meeting at each point add to exactly \(360^\circ\).
Polygon names and angle sums:
| Sides | Name | Angle sum |
|---|---|---|
| 3 | triangle | \(180^\circ\) |
| 4 | quadrilateral | \(360^\circ\) |
| 5 | pentagon | \(540^\circ\) |
| 6 | hexagon | \(720^\circ\) |
| 7 | heptagon | \(900^\circ\) |
| 8 | octagon | \(1080^\circ\) |
| 9 | nonagon | \(1260^\circ\) |
| 10 | decagon | \(1440^\circ\) |
| 12 | dodecagon | \(1800^\circ\) |
| \(n\) | \(n\)-gon | \((n - 2) \times 180^\circ\) |
How to find an angle in a polygon
- Count the sides \(n\) and work out \(S = (n - 2) \times 180\).
- Regular? Divide \(S\) by \(n\), or use exterior angle \(= 360 \div n\).
- Missing angle? Subtract the known angles from \(S\) and solve.
| \(S\) | \(=\) | \(180 \times (5 - 2)\) |
| \(\) | \(=\) | \(540^\circ\) |
\(540^\circ\)
| \(\theta\) | \(=\) | \(\dfrac{180 \times (5 - 2)}{5}\) |
| \(\) | \(=\) | \(\dfrac{540}{5} = 108^\circ\) |
\(108^\circ\)
| \(S\) | \(=\) | \((14 - 2) \times 180\) |
| \(\) | \(=\) | \(12 \times 180 = 2160^\circ\) |
\(2160^\circ\): the polygon splits into \(12\) triangles.
| \(S\) | \(=\) | \((6 - 2) \times 180 = 720\) |
| \(3c + 354\) | \(=\) | \(720\) |
| \(3c\) | \(=\) | \(366\) |
| \(c\) | \(=\) | \(122\) |
\(c = 122\). Check: \(354 + 3 \times 122 = 720\).
(i) each exterior angle
(ii) each interior angle
| \(360 \div 20\) | \(=\) | \(18\) |
| \(180 - 18\) | \(=\) | \(162\) |
| \((20 - 2) \times 180 \div 20\) | \(=\) | \(162\) |
(i) \(18^\circ\)
(ii) \(162^\circ\), which the interior sum confirms.
(i) the number of sides
(ii) each interior angle
(iii) whether tiles of this shape can cover a floor with no gaps
| \(n\) | \(=\) | \(360 \div 24 = 15\) |
| \(180 - 24\) | \(=\) | \(156\) |
| \(2 \times 156\) | \(=\) | \(312\) |
| \(3 \times 156\) | \(=\) | \(468\) |
(i) \(15\) sides
(ii) \(156^\circ\)
(iii) No: two tiles leave a \(48^\circ\) gap and three overlap.
| \(1620 \div 180\) | \(=\) | \(9\) |
| \(n\) | \(=\) | \(9 + 2 = 11\) |
\(11\) sides.
Common pitfalls
Frequently asked questions
What is the formula for the angle sum of a polygon?
\(S = (n - 2) \times 180^\circ\), where \(n\) is the number of sides.
How do I find one angle of a regular polygon?
Divide the angle sum by \(n\), or work out \(180 - 360 \div n\).
What do the exterior angles of a polygon add to?
\(360^\circ\) for any convex polygon.
How many sides does a regular polygon with 160 degree angles have?
Each exterior angle is \(20^\circ\), so it has \(360 \div 20 = 18\) sides.
Which regular polygons tile a floor?
Equilateral triangles, squares and regular hexagons, because \(60\), \(90\) and \(120\) divide exactly into \(360\).