Mathematics Year 7: Interior and exterior angles of a convex polygon
Mathematics, Year 7 (Portugal): Where the sums (n − 2) × 180° and 360° come from, and how to use them to find a missing angle.
Inside, outside and the two sums
In a convex polygon, the interior angle at a vertex lies inside the figure, between two consecutive sides. The exterior angle lies outside: it is the angle between one side and the extension of the other side at that vertex, and you take one per vertex. At the same vertex the two are supplementary: they add up to 180°. Draw all the diagonals from one vertex and a polygon with n sides splits into n − 2 triangles, so its interior angles add up to (n − 2) × 180°. The exterior angles always add up to 360°.
Where the two sums come from
Interior angles: the n − 2 triangles fill the whole polygon and each brings 180°, so the total is (n − 2) × 180°. Exterior angles: imagine walking around the polygon and turning, at each vertex, by the exterior angle. Back at the start you face the same way again: you have made one full turn, 360°, whether the polygon has 3 sides or 30. The two sums agree: since each interior angle is 180° minus the exterior one, the interior sum is n × 180° − 360°, which equals (n − 2) × 180°.
Finding a missing angle in a pentagon
A convex pentagon has four interior angles of 100°, 110°, 95° and 120°. Find the fifth. Sum of the interior angles: (5 − 2) × 180° = 3 × 180° = 540°. The four known ones add up to 100 + 110 + 95 + 120 = 425°, so the fifth measures 540 − 425 = 115°. Exterior angles: 180 − 100 = 80°, 70°, 85°, 60° and 180 − 115 = 65°; they add up to 80 + 70 + 85 + 60 + 65 = 360°. If it were regular, each interior angle would be 540 ÷ 5 = 108° and each exterior one 360 ÷ 5 = 72°.
The trap: n × 180°, and dividing by n
Two slips. One: writing n × 180° instead of (n − 2) × 180°. For the pentagon that gives 900° instead of 540°; always count the n − 2 triangles. Two: dividing the sum by the number of sides to get the angle of the polygon. That only works for regular polygons. In the pentagon above the angles are 100°, 110°, 95°, 120° and 115°, and none of them is 108°. And remember that the exterior angles do not depend on the number of sides: they add up to 360° in a triangle and in a dodecagon alike.
Where you see this
In block programming such as Scratch, you draw a regular polygon by repeating «move, then turn». What you turn by is the exterior angle, 360° ÷ n: 72° for a pentagon, 60° for a hexagon. On a floor tiled with regular hexagons, three corners meet at a point with no gaps, because 3 × 120° = 360°. And anyone drawing a plan or a part uses the sum to find an angle they cannot measure directly.
Test what you learned
At a vertex of a convex polygon, the interior angle measures 110°. How much does the exterior angle at that vertex measure?
- a) 20°
- b) 110°
- c) 70°
- d) 250°
Correct answer: c) 70° — Right: 180° − 110° = 70°. At the same vertex, interior and exterior angles are supplementary.
What do the interior angles of a convex polygon with 9 sides add up to?
- a) 1260°
- b) 1620°
- c) 1440°
- d) 1080°
Correct answer: a) 1260° — Right: 9 sides give 9 − 2 = 7 triangles, and 7 × 180° = 1260°.
Sara says: «A convex dodecagon has more sides than a triangle, so the sum of its exterior angles (one per vertex) is bigger.» What do you answer?
- a) She is right: more angles, so a bigger sum, just like the interior angles
- b) She is right only if the dodecagon is regular
- c) She is wrong: the exterior angles always add up to 180°
- d) She is wrong: in any convex polygon the exterior angles add up to 360°, whatever the number of sides
Correct answer: d) She is wrong: in any convex polygon the exterior angles add up to 360°, whatever the number of sides — Right: walking around any convex polygon you make one full turn, 360°, with 3 sides or with 12.
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