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Complete IGCSE Edexcel guide to polygon angle sums, regular polygon formulae, and finding the number of sides from angle information. Worked examples for grades 5-7.
The sum of interior angles of an n-sided polygon is (n - 2) x 180 degrees. Exterior angles of any convex polygon always sum to 360 degrees. For a regular polygon, each exterior angle equals 360 divided by n, and each interior angle equals 180 minus the exterior angle. To find the number of sides from a given interior angle, calculate the exterior angle first (180 minus interior), then divide 360 by the exterior angle.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
The interior angle sum of any polygon depends only on the number of sides. The formula is:
This formula works because every polygon can be divided into (n - 2) triangles by drawing diagonals from a single vertex, and each triangle has an angle sum of 180 degrees.
| Polygon | Sides (n) | Triangles (n-2) | Angle Sum |
|---|---|---|---|
| Triangle | 3 | 1 | 180 degrees |
| Quadrilateral | 4 | 2 | 360 degrees |
| Pentagon | 5 | 3 | 540 degrees |
| Hexagon | 6 | 4 | 720 degrees |
| Octagon | 8 | 6 | 1080 degrees |
| Decagon | 10 | 8 | 1440 degrees |
An exterior angle is formed by extending one side of the polygon at a vertex. At every vertex, the interior and exterior angles are supplementary (they add up to 180 degrees because they sit on a straight line).
The sum of all exterior angles of any convex polygon is always 360 degrees. You can visualise this by imagining walking around the perimeter: at each vertex you turn through the exterior angle, and by the time you complete the loop you have turned a full 360 degrees.
A regular polygon has all sides equal and all angles equal. This means every exterior angle is the same and every interior angle is the same.
Each exterior angle = 360 / n
Each interior angle = 180 - (360 / n)
Alternatively: Each interior angle = (n - 2) x 180 / n
Both interior angle formulae give the same result. The exterior angle method is generally faster and less prone to arithmetic errors in an exam.
This is the reverse problem and appears frequently on IGCSE papers. The method depends on which angle you are given.
Divide 360 by the exterior angle: n = 360 / exterior angle.
First find the exterior angle: exterior = 180 - interior. Then divide: n = 360 / exterior. For example, if the interior angle is 150 degrees, the exterior angle is 30 degrees, so n = 360 / 30 = 12 sides (a regular dodecagon).
🎯 Find the sum of interior angles of a hexagon.
Step 1 (Identify n): A hexagon has 6 sides, so n = 6.
Step 2 (Apply formula): Interior angle sum = (n - 2) x 180 = (6 - 2) x 180
Step 3 (Calculate): = 4 x 180 = 720 degrees
Answer: 720 degrees
🎯 Find each interior angle of a regular decagon.
Step 1 (Find exterior angle): A decagon has 10 sides. Each exterior angle = 360 / 10 = 36 degrees.
Step 2 (Find interior angle): Each interior angle = 180 - 36 = 144 degrees.
Step 3 (Verify): Interior angle sum = (10 - 2) x 180 = 1440 degrees. Each angle = 1440 / 10 = 144 degrees. Correct.
Answer: 144 degrees
🎯 Each interior angle of a regular polygon is 156 degrees. Find the number of sides.
Step 1 (Find exterior angle): Exterior angle = 180 - 156 = 24 degrees.
Step 2 (Find number of sides): n = 360 / exterior angle = 360 / 24 = 15.
Step 3 (State the answer): The polygon has 15 sides (a regular pentadecagon).
Answer: 15 sides
Confusing interior and exterior angles: The interior angle is inside the polygon. The exterior angle is formed by extending one side. They always add up to 180 degrees at each vertex. Mixing them up leads to completely wrong answers.
Using the wrong formula: The formula (n - 2) x 180 gives the total interior angle SUM, not a single interior angle. For a single angle of a regular polygon, divide the sum by n, or use the exterior angle method.
Dividing 360 by the interior angle: When finding the number of sides, you must divide 360 by the EXTERIOR angle, not the interior angle. This is the most frequent error on reverse polygon questions.
Assuming all polygons are regular: The exterior angle = 360/n formula only works for regular polygons where all angles are equal. Irregular polygons require a different approach.
Start with the exterior angle: For regular polygon questions, find the exterior angle first. Every other value (interior angle, number of sides, angle sum) follows directly from it.
Memorise the key angle sums: Triangle = 180, quadrilateral = 360, pentagon = 540, hexagon = 720. These save time on Paper 1 (non-calculator).
Check your answer is a whole number: The number of sides must be a positive integer. If you get a decimal, you have made an error somewhere. Go back and check.
Show your formula: Writing (n - 2) x 180 before substituting earns method marks even if the arithmetic goes wrong. Always show the formula first.
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