Maths · Suggested Grade 8
Class 8 Polygon Angles: Interior, Exterior & Missing Angles
A hexagon can be split into four triangles, so its interior angles total 720°. The six sides do not mean six triangles—and the total is different from one angle.
Original explanations and practice by BananaLearn · Published 31 October 2026
On a small screen, swipe across the illustration to read every label.
Start with the shape and the angle you are measuring
A polygon is a closed plane shape made of straight sides. A simple polygon has no crossing sides. This guide begins with convex polygons, whose interior angles are each less than 180°. A regular polygon has all sides equal and all interior angles equal. Equal sides alone are not enough to justify dividing an angle sum equally.
You should know that a triangle’s interior angles total 180° and angles on a straight line total 180°. Label an angle as interior or exterior before calculating. A diagram is a model of the stated relationships; do not infer equal angles just because a sketch looks balanced.
Why the interior sum is (n − 2) × 180°
Choose one vertex of a convex polygon and draw diagonals to all non-adjacent vertices. A pentagon becomes three triangles; a hexagon becomes four. In general, an n-sided polygon becomes n − 2 triangles. The angles of these triangles fill the polygon’s interior angles, giving a total of (n − 2) × 180°.
Count triangles, not diagonals or sides. From one vertex a hexagon needs three diagonals to form four triangles. If you instead join the centre to every vertex, you make six triangles, but their angle totals include a full 360° around the centre. Subtracting that 360° also gives 6 × 180° − 360° = 720°.
Worked example: an irregular pentagon
A pentagon has angles 95°, 110°, 105°, 120° and x°. The total is (5 − 2) × 180° = 540°. The four known angles add to 430°, so x = 540° − 430° = 110°. Nothing in the question says the pentagon is regular.
Dividing 540° by 5 would give 108°, which is the angle of a regular pentagon, not this one. Check your result by adding all five angles. It should return 540°. Also check that each angle fits the stated convex shape, rather than accepting a negative or reflex result without questioning the data.
An exterior angle records a turn
Extend one side at a vertex. The exterior angle used here is adjacent to the interior angle, so they add to 180°. If the interior angle is 135°, that exterior angle is 45°. Walk around a convex polygon, turning in the same direction at every vertex: the total turn is 360°.
For a regular polygon, all those exterior angles are equal, giving each exterior angle = 360° ÷ n. This full-turn sum applies when you take one consistent exterior turn at every vertex. Do not collect arbitrary angles outside a drawing and assume they total 360°.
Worked example: find the number of sides
A regular polygon has each exterior angle 24°. The number of sides is 360° ÷ 24° = 15. Each interior angle is 180° − 24° = 156°. Its interior sum is 15 × 156° = 2,340°, matching (15 − 2) × 180°.
If each interior angle is supplied instead, subtract it from 180° first. For 150°, the exterior angle is 30° and the polygon has 12 sides. A proposed regular polygon with exterior angle 50° would require 7.2 sides, so no such regular polygon exists. A polygon cannot have a fractional number of sides.
Worked example: an algebraic angle condition
In a convex quadrilateral the angles are x°, 2x°, 3x° and 4x°. Their total is 360°, so 10x = 360 and x = 36. The angles are 36°, 72°, 108° and 144°. All are below 180°, consistent with the convex condition.
The same ratio in a pentagon would not describe all its angles because it lists only four. Always check the number of angle expressions before building an equation. Write the final angles if asked for them; the intermediate value x alone may not answer the question.
When a useful rule needs a condition
The interior-sum formula also holds for simple concave polygons, but the convenient fan of diagonals from an arbitrary vertex may leave the shape. A suitable triangulation is needed. Self-crossing star figures require different angle conventions and are outside this beginner method.
For revision, write three lines: identify the polygon and whether it is regular; select total or single-angle relationship; substitute and check. A parent can ask why a calculation divides by n. The correct answer is that regularity makes all angles equal, not that every n-sided polygon shares one angle size.
A polygon angle decision sheet
Copy the record into a notebook or use it to organise your own answer. Blank spaces are for your work.
| Given information | Relationship | Calculation | Check |
|---|---|---|---|
| Convex polygon with n sides | Interior sum = (n − 2) × 180° | ___ | ___ |
| Regular polygon, one exterior angle | n = 360° ÷ exterior angle | ___ | Whole number of sides? |
| Regular polygon, one interior angle | Exterior = 180° − interior | ___ | ___ |
10 questions with explained answers
Try the questions before opening the answers. Explain the reason, then try a changed example to check your understanding.
Find the interior-angle sum of an octagon.
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1,080°
(8 − 2) × 180° = 1,080°.Find one interior angle of a regular hexagon.
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120°
Its sum is 720°; divide by 6.Find one exterior angle of a regular nonagon.
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40°
360° ÷ 9 = 40°.A regular polygon has exterior angle 18°. How many sides?
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20
360 ÷ 18 = 20.A regular polygon has interior angle 144°. How many sides?
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10
Exterior angle is 36°, so n = 360 ÷ 36.Four angles of a pentagon are 100°, 105°, 115° and 90°. Find the fifth.
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130°
540° − 410° = 130°.A quadrilateral has angles x°, x°, 2x°, 2x°. Find x.
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60°
6x = 360°.Can a regular polygon have exterior angle 50°?
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No
360 ÷ 50 is not a whole number of sides.How many triangles form a fan from one vertex of a convex heptagon?
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5
n − 2 = 7 − 2.Does an interior-angle sum of 720° prove the hexagon is regular?
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No
Every simple hexagon has that sum; equal individual angles and sides need further evidence.
Keep the ideas connected
- Review triangle angles
- Use supplementary angles
- Compare quadrilateral families
- Solve equations built from angles
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