Practice Maths

Angles in Polygons

Key Ideas

Key Terms

polygon
A closed flat shape with straight sides.
interior angle
The angle inside a polygon at each vertex.
exterior angle
The angle between one side of a polygon and the extension of the adjacent side; supplements the interior angle.
regular polygon
A polygon with all sides equal and all angles equal.
angle sum
The total of all interior angles; for n sides: S = (n − 2) × 180°.
vertex
A corner point where two sides of a polygon meet (plural: vertices).
• Interior angle sum: S = (n − 2) × 180°
• Each interior angle (regular): (n − 2) × 180° ÷ n
• Sum of exterior angles of ANY convex polygon = 360°
• Each exterior angle (regular): 360° ÷ n

Common Angle Sums

Polygon Sides (n) Interior angle sum Each angle (regular)
Triangle 3 180° 60°
Quadrilateral 4 360° 90°
Pentagon 5 540° 108°
Hexagon 6 720° 120°
Octagon 8 1080° 135°
Hot Tip The exterior angles of any convex polygon always add to 360° — no matter how many sides. Use this to find an unknown exterior angle or to work backwards to find n for a regular polygon.

Worked Example

Question: Find the size of each interior angle and each exterior angle of a regular decagon (10 sides).

Step 1 — Interior angle sum:
S = (10 − 2) × 180° = 8 × 180° = 1440°

Step 2 — Each interior angle:
Interior angle = 1440° ÷ 10 = 144°

Step 3 — Each exterior angle:
Exterior angle = 180° − 144° = 36°   (or 360° ÷ 10 = 36°)

Interior Angles of a Polygon

Any polygon can be divided into triangles by drawing diagonals from one vertex. A quadrilateral splits into 2 triangles, a pentagon into 3, a hexagon into 4, and so on. Since any triangle has an angle sum of 180°, and an n-sided polygon splits into (n − 2) triangles, the interior angle sum of any n-sided polygon is:

Interior angle sum = (n − 2) × 180°

Let's check some familiar shapes:

  • Triangle (n = 3): (3 − 2) × 180° = 1 × 180° = 180°. ✓
  • Quadrilateral (n = 4): (4 − 2) × 180° = 2 × 180° = 360°. ✓
  • Pentagon (n = 5): (5 − 2) × 180° = 3 × 180° = 540°.
  • Hexagon (n = 6): (6 − 2) × 180° = 4 × 180° = 720°.

Each Interior Angle of a Regular Polygon

A regular polygon has all sides equal and all angles equal. Since the total interior angle sum is (n − 2) × 180° and there are n equal angles, each interior angle is:

Each interior angle = (n − 2) × 180° ÷ n

Examples:

  • Regular pentagon: (5 − 2) × 180 ÷ 5 = 540 ÷ 5 = 108°.
  • Regular hexagon: (6 − 2) × 180 ÷ 6 = 720 ÷ 6 = 120°.
  • Regular octagon: (8 − 2) × 180 ÷ 8 = 1080 ÷ 8 = 135°.

This is why honeycomb cells are hexagonal — 120° angles fit together perfectly (three hexagons meeting at a vertex: 3 × 120° = 360°).

Exterior Angles

The exterior angle of a polygon is the angle between one side and the extension of the adjacent side. At each vertex, the interior angle and exterior angle form a straight line, so:

Interior angle + Exterior angle = 180°

An important fact: the sum of all exterior angles of any convex polygon is always 360°, regardless of the number of sides. Think of it this way: if you walked around the perimeter of any polygon, at each corner you would turn by the exterior angle. By the time you return to your starting point facing the same direction, you have turned through a total of exactly 360° — one full revolution.

For a regular polygon: each exterior angle = 360° ÷ n.

Finding a Missing Angle

These formulas can be used to find a missing angle inside a polygon when all other angles are known. Add up the known angles, then subtract from the total interior angle sum.

Example: A pentagon has four angles of 102°, 115°, 90°, and 128°. Find the fifth angle.
Total sum = (5 − 2) × 180 = 540°.
Sum of known angles = 102 + 115 + 90 + 128 = 435°.
Missing angle = 540 − 435 = 105°.

Finding the Number of Sides from an Angle

If you know each interior angle of a regular polygon, you can find n. Rearrange the formula:
(n − 2) × 180 = n × (interior angle). Expand and solve for n.
Alternatively, if you know the exterior angle: n = 360 ÷ exterior angle. (Simpler!)

Example: A regular polygon has each exterior angle = 24°. How many sides? n = 360 ÷ 24 = 15 sides.

Key tip: Memorise the exterior angle fact: all exterior angles add to 360°. It is often the quickest route to a solution. If a question gives you the interior angle of a regular polygon, convert to the exterior angle first (180 − interior), then find n = 360 ÷ exterior angle. This two-step method avoids solving a more complex equation.

Mastery Practice

  1. Calculate the interior angle sum for each polygon using S = (n − 2) × 180°. Fluency

    1. Pentagon (5 sides)
    2. Hexagon (6 sides)
    3. Heptagon (7 sides)
    4. Octagon (8 sides)
    5. Nonagon (9 sides)
    6. Decagon (10 sides)
    7. 12-sided polygon (dodecagon)
    8. 15-sided polygon

  2. For each regular polygon, find (i) each interior angle and (ii) each exterior angle. Fluency

    1. Equilateral triangle (3 sides)
    2. Square (4 sides)
    3. Regular pentagon (5 sides)
    4. Regular hexagon (6 sides)
    5. Regular octagon (8 sides)
    6. Regular decagon (10 sides)

  3. Find the unknown angle in each polygon. Fluency

    1. A pentagon has four angles: 90°, 110°, 130°, 100°. Find the fifth angle.
    2. A hexagon has five equal angles of 110°. Find the sixth angle.
    3. A quadrilateral has angles 3x, 4x, 5x, and 6x. Find x and all four angles.
    4. A triangle has angles (2a + 10)°, (3a − 5)°, and (a + 15)°. Find a and all three angles.
    5. An octagon has seven angles of 135° each. Find the eighth angle.

  4. Exterior angles of polygons. Understanding

    1. A regular polygon has exterior angles of 45°. How many sides does it have?
    2. A regular polygon has exterior angles of 24°. How many sides does it have?
    3. Each exterior angle of a regular polygon is 30°. Find the interior angle and the number of sides.
    4. A convex pentagon has exterior angles of 80°, 70°, 65°, and 55° at four vertices. Find the fifth exterior angle.
    5. Explain why the sum of exterior angles of any convex polygon is always 360°.

  5. Apply the angle sum rules to answer each question. Understanding

    1. A regular polygon has an interior angle of 150°. How many sides does it have?
    2. A regular polygon has an interior angle of 140°. How many sides does it have?
    3. Is it possible for a regular polygon to have each interior angle equal to 170°? Explain.
    4. ABCDE is a regular pentagon. AC is a diagonal. Find the angle ∠CAE (the angle at A between side AE and diagonal AC). Give a reason for each step.
    5. A polygon has an interior angle sum of 1800°. How many sides does it have?

  6. Angles in polygons problem solving. Problem Solving

    1. A tiler is using regular hexagonal tiles. Show that regular hexagons tessellate (tile without gaps) by showing that the angles around each meeting point sum to 360°.
    2. A regular polygon has 20 sides. A student claims each interior angle is 160° and each exterior angle is 20°. Verify whether the student is correct.
    3. In polygon ABCDEFG (regular heptagon, 7 sides): find the interior angle sum, each interior angle (to 1 decimal place), and each exterior angle (to 1 decimal place).
    4. The angles of a quadrilateral are in the ratio 2 : 3 : 4 : 6. Find all four angles and identify the most specific name for this quadrilateral if it has one pair of parallel sides.
  7. Polygon angles challenge. Use angle-in-polygon rules to solve each problem. Show all working.
    Problem Solving
    1. A polygon has an interior angle sum of 2520°. How many sides does it have? Name the polygon if it has a common name.
    2. Each interior angle of a regular polygon is five times its exterior angle. Find the number of sides.
    3. The interior angles of a hexagon are x°, (x + 10)°, (x + 20)°, (x + 30)°, (x + 40)°, and (x + 50)°. Find x and state all six angles.
    4. A regular polygon has each exterior angle equal to 18°. Find the number of sides, each interior angle, and the interior angle sum.
  8. Interior and exterior angles combined. Solve each multi-step problem with full working.
    Problem Solving
    1. Three regular polygons meet at a point without gaps or overlaps. Two of them are a regular hexagon and a regular square. What must the third polygon be? (Find the required angle at the point, then identify the polygon.)
    2. A convex polygon has n sides. Its interior angle sum is 4 times its exterior angle sum. Find n.
    3. The angles of a pentagon are in the ratio 2 : 3 : 4 : 5 : 6. Find all five angles.
    4. In a regular 12-sided polygon (dodecagon), two non-adjacent vertices are joined by a diagonal. The diagonal and two sides of the polygon form a triangle. Find all angles of this triangle.
  9. Polygon reasoning with algebra. Use angle-sum formulas and algebra to find unknowns.
    Problem Solving
    1. A regular polygon has interior angle = (4n − 8)° where n is the number of sides. Show that this expression is equivalent to the formula (n − 2) × 180 ÷ n and use it to find the polygon with interior angle 156°.
    2. An irregular polygon has n sides. All angles are equal to 160° except one, which is 120°. If the interior angle sum equals (n − 2) × 180°, find n.
    3. Two regular polygons have the same exterior angle size. The first has 6 sides and the second has 9 sides. Is this possible? If not, find the ratio of their exterior angles.
    4. A polygon has interior angles that form an arithmetic sequence starting at 100° with a common difference of 10°. The polygon has 8 sides. Find all angles and verify they satisfy the interior angle sum formula.
  10. Real-world polygon problems. Apply polygon angle rules to extended problems.
    Problem Solving
    1. A floor is tiled using regular octagons and squares that fit together without gaps. At each meeting point, one octagon and two squares meet. Verify this is correct by checking the angles at that meeting point sum to 360°.
    2. A pentagon ABCDE has ∠A = 100°, ∠B = 2∠E, ∠C = ∠D, and ∠B + ∠C = 250°. Find all five angles.
    3. The sum of the interior angles of polygon P is 1260° and the sum of the interior angles of polygon Q is 1620°. How many more sides does Q have than P? What are the two polygons?
    4. A star shape is formed by extending the sides of a regular pentagon. The five points of the star are triangles. Find the angle at each point of the star. (Hint: use the exterior angles of the pentagon.)
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