Exterior Angle of Polygon
In any convex polygon, the exterior angles add to \(360^\circ\), so each exterior angle of a regular polygon with \(n\) sides is \(\frac{360^\circ}{n}\). This NSW Year 9 Advanced maths subtopic uses this quick route to polygon angles (NESA outcome MA5‑GEO‑P‑01).
You will find exterior angles of regular polygons, the number of sides from an exterior or interior angle, and missing exterior angles of a pentagon or quadrilateral. You will explain why \(25^\circ\) is impossible, follow a robot that turns \(24^\circ\) at each corner, explain why hexagons tile a floor but pentagons do not, and prove the \(360^\circ\) result from the interior angle sum. It is part of Year 9 Advanced Properties of Geometric Figures.
Every question with a fully worked solution.
- Exterior Angle of Polygon - Video - Angles in polygons- exterior angles Watch
Theory
The exterior angles of a polygon always add to \(360^\circ\), whatever the number of sides. This Year 9 Advanced maths page uses that fact to find the exterior angle of a regular polygon, the number of sides from an interior angle, unknown angles written as expressions, and angles made when sides are produced.
An exterior angle of a polygon is the angle between one side and the extension of the side next to it. At every vertex the interior angle and the exterior angle lie on a straight line, so interior \(+\) exterior \(= 180^\circ\).
Walk once round a convex polygon. At each corner you turn through the exterior angle, and when you arrive back at the start facing the same way you have turned through one full turn. So the exterior angles add to \(360^\circ\) for every convex polygon: a triangle, a pentagon or a polygon with \(100\) sides.
In a regular polygon the exterior angles are all equal, so each one is \(\dfrac{360^\circ}{n}\). Turning this round, \(n = \dfrac{360}{\text{exterior angle}}\), which must be a whole number.
Exterior angles are often the quickest route through a polygon problem, because \(360\) is the same for every polygon and the fractions stay simple.
For any convex polygon, and for a regular polygon with \(n\) sides:
How to use exterior angles
- Given an interior angle of a regular polygon: exterior \(= 180^\circ -\) interior, then \(n = 360 \div\) exterior.
- Given \(n\): exterior \(= 360^\circ \div n\), then interior \(= 180^\circ -\) exterior.
- Exterior angles as expressions: add them, set the total equal to \(360\), solve, then substitute back.
- Interior linked to exterior ("\(k\) times" or "\(d^\circ\) more than"): let the exterior angle be \(e\), write the interior angle in terms of \(e\), and use interior \(+\) exterior \(= 180\).
- Groups of equal exterior angles: let one group have \(k\) angles, write the others in terms of \(k\), and set the total to \(360\). The number of sides is the total number of angles.
- Two regular polygons: write each exterior angle as \(\dfrac{360}{n}\) (or \(\dfrac{360}{n + 1}\), and so on), form one equation and multiply through by the denominators. If this leaves a product such as \(n(n + 1) = 56\), try consecutive whole numbers: \(7 \times 8 = 56\), so \(n = 7\).
- Sides produced to meet: the angles at the base of the new triangle are exterior angles of the polygon. Use the angle sum of a triangle.
- Turning problems: a robot or a walker turning the same angle at every corner traces a regular polygon, and the turn is the exterior angle.
- Tiling: copies of one regular polygon fit round a point with no gap only if its interior angle divides exactly into \(360^\circ\). Squares work: \(360 \div 90 = 4\) squares meet at each corner.
(i) the number of sides
(ii) the sum of its interior angles.
(i) Find the exterior angle first, then divide it into \(360^\circ\).
| \(e\) | \(=\) | \(180^\circ - 172^\circ = 8^\circ\) |
| \(n\) | \(=\) | \(\dfrac{360}{8}\) |
| \(\) | \(=\) | \(\mathbf{45}\) |
(ii)
| \(S\) | \(=\) | \(180(45 - 2)\) |
| \(\) | \(=\) | \(\mathbf{7740^\circ}\) |
Check: \(45 \times 172 = 7740\).
Let each exterior angle be \(e^\circ\), so each interior angle is \(7e^\circ\). They lie on a straight line.
| \(7e + e\) | \(=\) | \(180\) |
| \(8e\) | \(=\) | \(180\) |
| \(e\) | \(=\) | \(22.5\) |
| \(n\) | \(=\) | \(\dfrac{360}{22.5}\) |
| \(\) | \(=\) | \(\mathbf{16}\) |
Check: interior angle \(= 180 - 22.5 = 157.5^\circ\), and \(7 \times 22.5 = 157.5\).
The exterior angles of any convex polygon add to \(360^\circ\).
| \(x + (x + 15) + 2x + (2x - 15) + 3x\) | \(=\) | \(360\) |
| \(9x\) | \(=\) | \(360\) |
| \(x\) | \(=\) | \(40\) |
The exterior angles are \(40^\circ, 55^\circ, 80^\circ, 65^\circ\) and \(120^\circ\). The largest interior angle sits next to the smallest exterior angle.
| \(\text{largest interior}\) | \(=\) | \(180^\circ - 40^\circ\) |
| \(\) | \(=\) | \(\mathbf{140^\circ}\) |
Check: the interior angles \(140, 125, 100, 115, 60\) add to \(540^\circ\), the angle sum of a pentagon.
\(\angle PBC\) is the exterior angle at \(B\) (\(AB\) produced), and \(\angle PCB\) is the exterior angle at \(C\).
| \(\angle PBC = \angle PCB\) | \(=\) | \(\dfrac{360^\circ}{8} = 45^\circ\) |
| \(\angle BPC\) | \(=\) | \(180^\circ - 45^\circ - 45^\circ\) |
| \(\) | \(=\) | \(\mathbf{90^\circ}\) |
(angle sum of a triangle). Two angles are equal, so \(\triangle BPC\) is a right-angled isosceles triangle with \(PB = PC\).
Common pitfalls
Frequently asked questions
What do the exterior angles of a polygon add up to?
The exterior angles of any convex polygon add up to \(360^\circ\), whatever the number of sides. Walking once round the polygon you turn through each exterior angle and make one full turn.
How do you find the exterior angle of a regular polygon?
Divide \(360^\circ\) by the number of sides. A regular polygon with \(10\) sides has exterior angles of \(\dfrac{360^\circ}{10} = 36^\circ\).
How do you find the number of sides from an interior angle?
Subtract the interior angle from \(180^\circ\) to get the exterior angle, then divide \(360\) by the exterior angle. The answer must be a whole number.
What is the difference between interior and exterior angles?
An interior angle is inside the polygon between two sides. An exterior angle is outside, between one side and the next side extended. At each vertex they add to \(180^\circ\).
Can a regular polygon have an exterior angle of any size?
No. The exterior angle must divide exactly into \(360^\circ\), and it can be at most \(120^\circ\), for an equilateral triangle. For example \(50^\circ\) is impossible because \(360 \div 50 = 7.2\).