Polygon angles
Use interior totals for any simple polygon. Equal regular angles share those totals; convex exterior turning angles make one full turn
- Interior total A simple n-sided polygon can be divided into n − 2 triangles. This works for convex and concave polygons \(\displaystyle S=180(n-2)\)
- Missing angle Add the known interior angles, then subtract that total from S. Keep a reflex interior angle greater than 180° \(\displaystyle x=S-\text{known total}\)
- Regular interior angle Regular means all sides and all interior angles are equal. Divide S equally between the n angles \(\displaystyle I=S\div n\)
- Exterior angle The exterior turning angles, one at each vertex of a convex polygon, total 360°. A regular polygon shares this turn equally \(\displaystyle E=360^\circ\div n\)
- Interior and exterior In a convex polygon, an interior angle and its adjacent exterior angle form a straight line \(\displaystyle I+E=180^\circ\)
- Regular perimeter Multiply the common side length s by n. Angles use degrees; perimeter uses length units \(\displaystyle P=ns\)
- Why there are n−2 triangles For a convex polygon, a fan from one vertex uses n−3 diagonals and creates n−2 triangles. Their angles recombine into the polygon interior angles. Concave polygons also triangulate, but the diagonals may need different starting vertices
- Why the exterior total is 360 degrees Walk once around a convex polygon in one direction. The turn at each corner is its exterior angle. After a complete circuit, the total turn is one full rotation. Equal turns require equal interior angles
- Work backwards to a side count From a regular exterior angle E, divide 360 by E. From a regular interior angle I, first find E=180−I. A polygon must have a whole number of sides, at least 3. \(\displaystyle n=\frac{360}{E}\)
Worked example
A regular hexagon has side length 9 cm. Find each interior angle and the perimeter
- \(\displaystyle S=180(6-2)=720^\circ\)The six-sided polygon has an interior-angle total of 720 degrees
- \(\displaystyle I=\frac{720^\circ}{6}=120^\circ\)Equal regular angles share the total equally, so each interior angle is 120 degrees
- \(\displaystyle P=6\times9\ \mathrm{cm}=54\ \mathrm{cm}\)The perimeter is six equal sides of 9 cm, giving 54 cm
Common mistakes
Difficulty levels
Questions about this topic
Why does the polygon angle formula use n - 2?+
A convex polygon can be split into n − 2 triangles by diagonals from one vertex. A concave polygon can also be triangulated using diagonals inside the shape, although they may need to start at different vertices
How are regular interior and exterior angles related?+
A regular interior angle and its exterior angle on a straight line add to 180 degrees. The exterior angle is also 360 divided by the number of sides
How is the perimeter of a regular polygon found?+
Multiply the common side length by the number of sides
Can I use a single fan of diagonals for every concave polygon?+
No. Some diagonals may lie outside the shape. Use an interior triangulation, allowing diagonals from different vertices. The n−2 triangle count still holds for a simple polygon
How do I find the side count from an interior angle?+
For a regular polygon, subtract the interior angle from 180 degrees to get the exterior angle, then divide 360 by that exterior angle. The result must be an integer at least 3.
Are equal angles enough to make a polygon regular?+
No. All sides must also be equal. A non-square rectangle is equiangular but not regular. Its equal angles can still be found by dividing its angle total equally
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 7: Geometry and similarity