Polygon angles
Use interior totals for any simple polygon. Equal regular angles share those totals; convex exterior turning angles make one full turn
\[S=180(n-2)\]
- 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 \(\displaystyle P=ns\)
Worked example
A regular hexagon has side length 9 cm. Find each interior angle and the perimeter
\[S=180(6-2)\]
- \(\displaystyle S=180(6-2)=720^\circ\)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
Using the number of sides alone
✕\(\displaystyle S=180(5)=900^\circ\)✓\(\displaystyle S=180(5-2)=540^\circ\)
The interior-angle total comes from n - 2 triangles, so using n instead of n - 2 gives a total that is too largeNot dividing a regular angle sum
✕\(\displaystyle I=720^\circ\)✓\(\displaystyle I=720^\circ\div6=120^\circ\)
The total is shared by all six equal interior angles, so it must be divided by 6Confusing interior and exterior angles
✕\(\displaystyle E=135^\circ\)✓\(\displaystyle E=360^\circ\div8=45^\circ\)
The exterior angles of a regular polygon share a full turn of 360 degrees, not the interior angle valueDifficulty levels
introName a polygon and find its interior-angle sum
standardFind a missing convex interior angle or one regular interior angle
challengeHandle a known reflex angle, a regular exterior angle, or interior angle with perimeter
examSelect the appropriate rule and calculate the requested angle or perimeter
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 calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 7: Geometry and similarity
In this chapter
Angle facts and parallel linesTriangle angle sum, special triangles and the exterior angle theoremQuadrilateral angle sums and special-quadrilateral propertiesPolygon angle sums and regular polygonsCongruent triangle testsProve congruence and use it to prove a further resultEnlargement transformation and scale factor7G Enlargement and scale: Similar figures and scale factorsTests for similar triangles7H Similar triangle tests: Similar figures and scale factorsApplied similarity and scaleSimilar solids and scaling volumeDesign, test and refine geometrical algorithms