Interior and Exterior Angles of Polygons

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The interior angles of a polygon with n sides add up to (n - 2) \cdot 180^\circ. The exterior angles, one at each vertex, always add up to 360°. In a regular polygon, all angles are equal, so each interior angle is the sum divided by n, and each exterior angle is 360^\circ \div n.

Before this lesson: You should know the Triangle Angle Sum Theorem, Angle Pairs and Measuring and Classifying Angles.

Polygons and Their Angles

A polygon is a closed flat figure made of straight sides. A polygon with n sides also has n vertices and n interior angles. In a regular polygon, every side is the same length and every angle is the same size. A convex polygon has no dents: every interior angle is less than 180°.

Here is the key idea. Pick one vertex and draw every diagonal you can from it. The diagonals cut the polygon into triangles. There are always 2 fewer triangles than sides, and each triangle’s angles add to 180°.

A hexagon with vertices labeled A through F. Three dashed orange diagonals are drawn from vertex A to C, D and E, splitting the hexagon into 4 triangles, numbered 1 to 4. A note reads 6 sides, 4 triangles, 4 times 180 degrees equals 720 degrees.
Diagonals from one vertex split a hexagon into 4 triangles, so its angles add to 4 × 180° = 720°.

Polygon Angle Formulas

What you want Formula
Sum of the interior angles S = (n - 2) \cdot 180^\circ
One interior angle of a regular polygon \dfrac{(n - 2) \cdot 180^\circ}{n}
Sum of the exterior angles (one per vertex) 360^\circ
One exterior angle of a regular polygon \dfrac{360^\circ}{n}
Interior and exterior angle at the same vertex They add to 180^\circ

An exterior angle is formed by one side and the extension of the next side. It forms a linear pair with the interior angle at that vertex. Walk all the way around a polygon, turning at each corner: you make one full turn, 360°. That is why the exterior angles always add to 360°, no matter how many sides there are.

A pentagon with each side extended past one vertex, going around in the same direction. At each vertex, the exterior angle between the extended side and the next side is marked with an orange arc and labeled 1 through 5. A note reads the exterior angles add to 360 degrees, one full turn.
Walking around the pentagon, the five turns make one full circle, so the exterior angles add to 360°.

How to Find Polygon Angles

  1. Count the sides, n.
  2. For the interior angle sum, subtract 2 and multiply by 180°.
  3. For a regular polygon, divide the sum by n to get one interior angle.
  4. For one exterior angle of a regular polygon, divide 360° by n.
  5. To find n from a regular polygon’s angle, find the exterior angle and divide it into 360°.

Worked Examples

Example 1: Interior angle sums

Find the sum of the interior angles of a hexagon (6 sides) and of an octagon (8 sides).

  1. Hexagon: subtract 2 from the number of sides: 6 - 2 = 4
  2. Multiply by 180°: 4 \times 180^\circ = 720^\circ
  3. Octagon: subtract 2: 8 - 2 = 6
  4. Multiply by 180°: 6 \times 180^\circ = 1080^\circ

Answer: A hexagon’s angles add to 720°, and an octagon’s add to 1080°.

Example 2: One angle of a regular polygon

Find each interior angle and each exterior angle of a regular decagon (10 sides).

  1. Find the sum: (10 - 2) \times 180^\circ = 8 \times 180^\circ = 1440^\circ
  2. Divide by 10 for one interior angle: 1440^\circ \div 10 = 144^\circ
  3. Divide 360° by 10 for one exterior angle: 360^\circ \div 10 = 36^\circ
  4. Check: the interior and exterior angles add to 180°: 144^\circ + 36^\circ = 180^\circ

Answer: Each interior angle is 144°, and each exterior angle is 36°.

Example 3: Find the number of sides

Each interior angle of a regular polygon measures 156°. How many sides does it have?

  1. The interior and exterior angles at a vertex add to 180°, so subtract: 180^\circ - 156^\circ = 24^\circ
  2. The exterior angles add to 360°, and each is 24°. Divide: 360^\circ \div 24^\circ = 15
  3. Check with the sum formula: (15 - 2) \times 180^\circ = 13 \times 180^\circ = 2340^\circ, and 2340^\circ \div 15 = 156^\circ.

Answer: The polygon has 15 sides.

Example 4: A missing angle

A pentagon has angles of 100°, 110°, 95°, 120° and x°. Find x.

  1. Find the angle sum of a pentagon: (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
  2. Add the four known angles: 100 + 110 + 95 + 120 = 425
  3. Subtract from the sum: 540 - 425 = 115
  4. Check: 425 + 115 = 540

Answer: x = 115.

Example 5: Word problem with floor tiles

A tile store sells regular hexagons and regular pentagons. To cover a floor with no gaps, the angles that meet at each corner must add to exactly 360°. Which shape works?

  1. Hexagon: one interior angle is \tfrac{(6 - 2) \times 180^\circ}{6} = \tfrac{720^\circ}{6} = 120^\circ
  2. Divide into 360°: 360^\circ \div 120^\circ = 3. Exactly 3 hexagons fit around a point.
  3. Pentagon: one interior angle is \tfrac{(5 - 2) \times 180^\circ}{5} = \tfrac{540^\circ}{5} = 108^\circ
  4. Divide into 360°: 360 \div 108 = 3.\overline{3}. That is not a whole number, so pentagons leave gaps.

Answer: The regular hexagons work, three at each corner. Regular pentagons can’t cover the floor without gaps.

Common Mistakes

Mistake Example of the error Fix
Multiplying n by 180° Hexagon sum: 6 \times 180^\circ = 1080^\circ Subtract 2 first: (6 - 2) \times 180^\circ = 720^\circ.
Thinking the exterior sum grows with n “An octagon’s exterior angles add to 720°” The exterior angles of any convex polygon add to 360°.
Dividing the sum when the polygon isn’t regular Saying each angle of an irregular pentagon is 108° Only regular polygons have equal angles.
Using the interior angle to find n 360 \div 156 Use the exterior angle: 180 - 156 = 24, then 360 \div 24 = 15.
Counting vertices wrong Calling an 8-sided figure a hexagon Hexagon: 6 sides. Heptagon: 7. Octagon: 8. Nonagon: 9. Decagon: 10.

Angle Measures of Common Regular Polygons

Polygon Sides Interior sum Each interior angle Each exterior angle
Triangle 3 180° 60° 120°
Square 4 360° 90° 90°
Pentagon 5 540° 108° 72°
Hexagon 6 720° 120° 60°
Octagon 8 1080° 135° 45°
Decagon 10 1440° 144° 36°

A stop sign is a regular octagon, so each of its corners measures 135°. The 3-side row matches the equilateral triangle, with 60° angles.

Practice Problems

  1. Find the sum of the interior angles of a nonagon (9 sides).
  2. Find the sum of the interior angles of a 12-sided polygon.
  3. Find the measure of each interior angle of a regular pentagon.
  4. Find each exterior angle and each interior angle of a regular 12-sided polygon.
  5. Each exterior angle of a regular polygon is 40°. How many sides does it have?
  6. Each interior angle of a regular polygon is 162°. How many sides does it have?
  7. A quadrilateral has angles of 85°, 95°, 110° and x°. Find x.
  8. The exterior angles of a quadrilateral are x°, 2x°, 3x° and 4x°. Find x and each exterior angle.
  9. The interior angles of a polygon add to 2340°. How many sides does it have?
  10. Can a regular polygon have an interior angle of 130°? Explain.
Show answers
  1. Subtract 2: 9 - 2 = 7
    Multiply: 7 \times 180 = 1260
    The sum is 1260°.
  2. Subtract 2: 12 - 2 = 10
    Multiply: 10 \times 180 = 1800
    The sum is 1800°.
  3. Sum: (5 - 2) \times 180 = 3 \times 180 = 540
    Divide by 5: 540 \div 5 = 108
    Each interior angle is 108°.
  4. Exterior: 360 \div 12 = 30, so 30°
    Interior: 180 - 30 = 150, so 150°
  5. Divide: 360 \div 40 = 9
    The polygon has 9 sides.
  6. Exterior angle: 180 - 162 = 18
    Divide: 360 \div 18 = 20
    The polygon has 20 sides.
  7. A quadrilateral’s angles add to (4 - 2) \times 180 = 360
    Add the known angles: 85 + 95 + 110 = 290
    Subtract: 360 - 290 = 70
    x = 70
  8. The exterior angles add to 360°: x + 2x + 3x + 4x = 360
    Combine like terms (see combining like terms): 10x = 360
    Divide by 10: x = 36
    The exterior angles are 36°, 72°, 108° and 144°.
  9. Set the formula equal to the sum: (n - 2) \times 180 = 2340
    Divide both sides by 180: n - 2 = 13
    Add 2: n = 15
    The polygon has 15 sides.
  10. Exterior angle: 180 - 130 = 50
    Divide: 360 \div 50 = 7.2
    The number of sides must be a whole number, so no regular polygon has a 130° angle.

FAQ

What is the formula for the sum of interior angles of a polygon?

The sum is (n - 2) \cdot 180^\circ, where n is the number of sides. For example, a pentagon’s angles add to 3 \cdot 180^\circ = 540^\circ.

Why do exterior angles always add up to 360°?

If you walk around the polygon and turn at each corner, you end up facing the way you started. That is one full turn, which is 360°.

How do you find one interior angle of a regular polygon?

Find the interior angle sum with (n - 2) \cdot 180^\circ, then divide by n. Or find the exterior angle, 360^\circ \div n, and subtract it from 180°.

How many sides does a polygon have if its angles add to 1080°?

Solve (n - 2) \cdot 180 = 1080. Divide by 180 to get n - 2 = 6, so n = 8. It is an octagon.

Summary

  • Interior angle sum: (n - 2) \cdot 180^\circ.
  • One interior angle of a regular polygon: the sum divided by n.
  • The exterior angles of any convex polygon add to 360°.
  • One exterior angle of a regular polygon: 360^\circ \div n.
  • An interior angle and its exterior angle add to 180°.
  • To find n, divide 360° by one exterior angle.

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