Interior Angle Sum of a Polygon

A polygon's interior angles are the angles inside its boundary. Their total depends only on the number of sides, not on whether the polygon is regular or irregular, convex or concave, as long as it is a simple polygon whose sides do not cross. The key formula is S = (n − 2) × 180°, where n is the number of sides and S is the sum of the interior angles.

Why the formula works

Choose one vertex and draw diagonals to the nonadjacent vertices. The polygon is split into triangles that do not overlap. An n-sided polygon creates n − 2 triangles: a triangle creates one, a quadrilateral creates two, a pentagon creates three, and a hexagon creates four. Each triangle has an angle sum of 180°, so the polygon's interior-angle sum is (n − 2) groups of 180°.

For example, a pentagon has five sides, so its sum is (5 − 2) × 180° = 540°. A hexagon has six sides: (6 − 2) × 180° = 720°. The result is a total, not the measure of each angle unless the polygon is regular and the angles are equal.

Finding each angle in a regular polygon

A regular polygon has equal sides and equal interior angles. Divide the total by the number of angles: each interior angle = [(n − 2) × 180°] ÷ n. A regular pentagon has each angle 540° ÷ 5 = 108°. A regular octagon has total (8 − 2) × 180° = 1,080°, so each interior angle is 1,080° ÷ 8 = 135°. Do not divide by n until the problem asks for each equal angle.

Working backward from the sum

If a problem gives the interior-angle total, solve S = (n − 2) × 180° for n. Divide the total by 180°, then add 2. If the total is 1,260°, then 1,260° ÷ 180° = 7, and n = 7 + 2 = 9 sides. The result should be a whole number at least 3. A nonwhole result usually means the given total is not the exact interior-angle sum of a polygon.

Finding a missing angle

For an irregular polygon, use the total and subtract the known angles. A quadrilateral's angles total (4 − 2) × 180° = 360°. If three angles are 85°, 95°, and 110°, the fourth is 360° − 85° − 95° − 110° = 70°. The known angles do not need to be equal. Check that the missing angle is a plausible interior angle for the shape shown.

Some questions describe angles with variables. For a pentagon with angle measures x, x + 10°, 2x, 100°, and 110°, set their sum equal to 540°: x + (x + 10) + 2x + 100 + 110 = 540. Combine terms to get 4x + 220 = 540, so x = 80°. Substitute back into every expression and confirm the angles total 540°.

Interior and exterior angles

A common mix-up is using an exterior-angle fact for an interior-angle question. For a convex polygon, one exterior angle at each vertex sums to 360°. That is different from the interior-angle total, which grows with the number of sides. At each vertex, an interior angle and its adjacent exterior angle form a linear pair and sum to 180°. Thus a regular polygon's exterior angle is 360° ÷ n, while its interior angle is 180° − 360° ÷ n.

For a regular decagon, each exterior angle is 360° ÷ 10 = 36°, so each interior angle is 180° − 36° = 144°. This agrees with the interior-sum method: (10 − 2) × 180° ÷ 10 = 144°. The two methods can check each other when the polygon is regular.

You can work backward from the measure of one regular polygon's interior angle. First find the exterior angle by subtracting from 180°, then divide 360° by that exterior angle. If each interior angle is 150°, the exterior angle is 30°, so n = 360° ÷ 30° = 12 sides. Check with the interior-angle formula: (12 − 2) × 180° ÷ 12 = 150°. If the calculation does not give a whole number of sides, the stated angle cannot belong to a regular polygon.

An irregular polygon's exterior angles still total 360° when one consistently directed exterior angle is taken at each vertex of a simple polygon. Individual turns may vary, so you cannot divide by n to find each exterior angle unless the polygon is regular. For a concave polygon, one interior angle is reflex (greater than 180°); the triangulation formula still gives the interior sum, but do not assume every interior angle is acute or obtuse based on a convex sketch.

Common errors and a fast check

• Count sides, not vertices in a drawing that hides one corner; a polygon has the same number of each, but the count must be accurate.
• Subtract 2 from n before multiplying by 180°.
• Treat the formula as a sum unless the polygon is regular or the question states the angles are equal.
• Keep angle units consistent; if measures are in degrees, write the total in degrees.
• For a triangle, the formula gives (3 − 2) × 180° = 180°, a useful check.
• For a quadrilateral, it gives 360°, another familiar check.

An irregular polygon can have one very large interior angle and several smaller ones; the total formula still applies. A concave polygon may have a reflex interior angle greater than 180°, but the same triangulation reasoning accounts for it. Follow the diagram and definitions given in the problem, and do not assume regularity from a neat-looking sketch.

Exam takeaway

Count n sides and calculate (n − 2) × 180° for the total. Divide by n only when equal angles are guaranteed, as in a regular polygon. When one angle is missing, subtract the known measures from the total; when the sum is given, reverse the formula to find the number of sides.

Why the polygon formula works

Choose one vertex of a convex polygon and draw diagonals from it to every nonadjacent vertex. The diagonals divide an n-sided polygon into n − 2 triangles. Each triangle has an angle sum of 180°, so the polygon’s interior-angle sum is (n − 2) × 180°. A pentagon becomes three triangles and has a sum of 540°; a decagon becomes eight triangles and has a sum of 1,440°.

Find a missing interior angle

The formula gives the total, not each individual angle unless the polygon is regular. For a pentagon whose four known interior angles are 100°, 110°, 120°, and 130°, the total is 540°. The missing angle is 540° − 460° = 80°. Check that the result is consistent with the polygon’s shape and that all five angles were counted exactly once.

Work backward to find the number of sides

If the interior-angle sum is given, solve S = (n − 2)180 for n: n = S ÷ 180 + 2. A sum of 1,080° gives n = 1,080 ÷ 180 + 2 = 8, so the polygon is an octagon. A non-integer result is a signal to revisit the stated sum or the setup, because a polygon cannot have a fractional number of sides.

Regular polygons and exterior angles

In a regular polygon, all interior angles are equal, so divide the sum by n to find one angle. A regular hexagon has sum (6 − 2)180 = 720°, and each interior angle is 120°. The exterior angle formed by extending one side is supplementary to that interior angle, giving 60°. The exterior angles, one at each vertex and measured in the same direction, add to 360°; this provides a separate check.

Do not assume equal angles just because the figure has equal side marks unless the problem identifies it as regular or supplies enough information. The sum formula applies to a simple convex polygon regardless of whether its sides and angles are equal. For a concave polygon, the same interior-angle sum holds, though a diagram may make the angle measurements less obvious.