What Are Polygons?

A polygon is a 2-dimensional shape with straight sides. Examples of polygons include triangles, quadrilaterals, pentagons, hexagons, and so on. Each polygon is classified based on the number of sides it has.

Interior Angles of a Polygon

The interior angles of a polygon are the angles found inside the shape. One of the key properties of polygons is that the sum of the interior angles depends on the number of sides the polygon has.

Formula for the Sum of Interior Angles

To find the sum of the interior angles of a polygon, you can use the formula:
$$
\text{Sum of interior angles} = (n – 2) \times 180^\circ
$$
Where:

  • $$n$$ is the number of sides of the polygon.
  • Each time you add a side to a polygon, you add an additional triangle to the shape, which is why the sum increases by $$180^\circ$$ for every extra side.

For example:

  • A triangle (3 sides) has a sum of interior angles of:
    $$
    (3 – 2) \times 180^\circ = 180^\circ
    $$
  • A quadrilateral (4 sides) has a sum of interior angles of:
    $$
    (4 – 2) \times 180^\circ = 360^\circ
    $$
  • A pentagon (5 sides) has a sum of interior angles of:
    $$
    (5 – 2) \times 180^\circ = 540^\circ
    $$

Exterior Angles of a Polygon

The exterior angles of a polygon are the angles formed between one side of the polygon and the extension of an adjacent side. One important fact about the exterior angles of any polygon is that they always sum to $$360^\circ$$, regardless of the number of sides.

Practice Questions: Sums of Angles in Polygons

Easy Level Questions

  1. What is the sum of the interior angles of a triangle?
  2. What is the sum of the interior angles of a quadrilateral?
  3. How many degrees are in each interior angle of an equilateral triangle?
  4. What is the sum of the exterior angles of any polygon?
  5. What is the sum of the interior angles of a pentagon?
  6. How many degrees are in each interior angle of a square?
  7. What is the sum of the interior angles of a hexagon?
  8. How many degrees are in each interior angle of a regular pentagon?
  9. What is the sum of the interior angles of a heptagon (7-sided polygon)?
  10. What is the sum of the interior angles of an octagon (8-sided polygon)?
  11. How many degrees are in each interior angle of a regular hexagon?
  12. What is the sum of the interior angles of a nonagon (9-sided polygon)?
  13. How many degrees are in each interior angle of a regular octagon?
  14. What is the sum of the interior angles of a decagon (10-sided polygon)?
  15. How many degrees are in each interior angle of a regular decagon?
  16. What is the sum of the interior angles of a dodecagon (12-sided polygon)?
  17. How many degrees are in each interior angle of a regular dodecagon?
  18. What is the sum of the interior angles of a polygon with 6 sides?
  19. What is the sum of the interior angles of a polygon with 10 sides?
  20. What is the sum of the interior angles of a polygon with 4 sides?

Medium Level Questions

  1. Find the sum of the interior angles of a polygon with 7 sides.
  2. What is the sum of the interior angles of a polygon with 9 sides?
  3. What is the sum of the interior angles of a polygon with 11 sides?
  4. Find the sum of the interior angles of a polygon with 15 sides.
  5. What is the measure of each interior angle of a regular pentagon?
  6. How many degrees are in each interior angle of a regular heptagon?
  7. What is the sum of the interior angles of a 20-sided polygon?
  8. What is the measure of each interior angle of a regular nonagon?
  9. What is the measure of each exterior angle of a regular hexagon?
  10. Find the sum of the exterior angles of a polygon with 12 sides.
  11. What is the sum of the exterior angles of a polygon with 15 sides?
  12. What is the measure of each interior angle of a regular dodecagon?
  13. Find the sum of the interior angles of a polygon with 18 sides.
  14. What is the measure of each interior angle of a regular 20-sided polygon?
  15. What is the measure of each exterior angle of a regular octagon?
  16. Find the sum of the interior angles of a polygon with 25 sides.
  17. What is the sum of the interior angles of a polygon with 30 sides?
  18. Find the sum of the exterior angles of a polygon with 8 sides.
  19. What is the measure of each interior angle of a regular heptagon?
  20. How many sides does a polygon have if the sum of its interior angles is $$1080^\circ$$?

Hard Level Questions

  1. What is the measure of each interior angle of a regular 24-sided polygon?
  2. What is the sum of the interior angles of a polygon with 40 sides?
  3. How many sides does a polygon have if each exterior angle measures $$15^\circ$$?
  4. Find the sum of the interior angles of a polygon with 50 sides.
  5. What is the measure of each exterior angle of a regular 30-sided polygon?
  6. How many sides does a polygon have if the sum of its interior angles is $$1440^\circ$$?
  7. What is the sum of the exterior angles of a polygon with 20 sides?
  8. How many sides does a polygon have if the sum of its exterior angles is $$180^\circ$$?
  9. What is the measure of each interior angle of a regular 18-sided polygon?
  10. Find the sum of the interior angles of a polygon with 60 sides.
  11. What is the measure of each interior angle of a regular 36-sided polygon?
  12. How many sides does a polygon have if each exterior angle is $$9^\circ$$?
  13. What is the sum of the exterior angles of a polygon with 50 sides?
  14. What is the measure of each exterior angle of a regular 24-sided polygon?
  15. Find the sum of the interior angles of a polygon with 100 sides.
  16. What is the measure of each interior angle of a regular 50-sided polygon?
  17. How many sides does a polygon have if each interior angle measures $$150^\circ$$?
  18. Find the sum of the exterior angles of a regular 36-sided polygon.
  19. What is the measure of each exterior angle of a regular polygon with 72 sides?
  20. How many sides does a polygon have if each interior angle measures $$160^\circ$$?

Answers with Explanation

Easy Level Answers

  1. Answer: $$180^\circ$$
    Explanation: A triangle has 3 sides. Using the formula for the sum of interior angles:
    $$ (3 – 2) \times 180^\circ = 180^\circ $$
  2. Answer: $$360^\circ$$
    Explanation: A quadrilateral has 4 sides:
    $$ (4 – 2) \times 180^\circ = 360^\circ $$
  3. Answer: $$60^\circ$$
    Explanation: An equilateral triangle has 3 equal interior angles, and the sum of the interior angles is $$180^\circ$$, so each angle is:
    $$ \frac{180^\circ}{3} = 60^\circ $$
  4. Answer: $$360^\circ$$
    Explanation: The sum of the exterior angles of any polygon is always $$360^\circ$$.
  5. Answer: $$540^\circ$$
    Explanation: A pentagon has 5 sides:
    $$ (5 – 2) \times 180^\circ = 540^\circ $$
  6. Answer: $$90^\circ$$
    Explanation: A square is a regular quadrilateral, so each angle is:
    $$ \frac{360^\circ}{4} = 90^\circ $$
  7. Answer: $$720^\circ$$
    Explanation: A hexagon has 6 sides:
    $$ (6 – 2) \times 180^\circ = 720^\circ $$
  8. Answer: $$108^\circ$$
    Explanation: A regular pentagon has 5 equal angles. The sum of the interior angles is $$540^\circ$$, so each angle is:
    $$ \frac{540^\circ}{5} = 108^\circ $$
  9. Answer: $$900^\circ$$
    Explanation: A heptagon has 7 sides:
    $$ (7 – 2) \times 180^\circ = 900^\circ $$
  10. Answer: $$1080^\circ$$
    Explanation: An octagon has 8 sides:
    $$ (8 – 2) \times 180^\circ = 1080^\circ $$
  11. Answer: $$120^\circ$$
    Explanation: A regular hexagon has 6 equal angles, so each interior angle is:
    $$ \frac{720^\circ}{6} = 120^\

circ $$

  1. Answer: $$1260^\circ$$
    Explanation: A nonagon has 9 sides:
    $$ (9 – 2) \times 180^\circ = 1260^\circ $$
  2. Answer: $$135^\circ$$
    Explanation: A regular octagon has 8 equal angles, so each interior angle is:
    $$ \frac{1080^\circ}{8} = 135^\circ $$
  3. Answer: $$1440^\circ$$
    Explanation: A decagon has 10 sides:
    $$ (10 – 2) \times 180^\circ = 1440^\circ $$
  4. Answer: $$144^\circ$$
    Explanation: A regular decagon has 10 equal angles, so each interior angle is:
    $$ \frac{1440^\circ}{10} = 144^\circ $$
  5. Answer: $$1800^\circ$$
    Explanation: A dodecagon has 12 sides:
    $$ (12 – 2) \times 180^\circ = 1800^\circ $$
  6. Answer: $$150^\circ$$
    Explanation: A regular dodecagon has 12 equal angles, so each interior angle is:
    $$ \frac{1800^\circ}{12} = 150^\circ $$
  7. Answer: $$720^\circ$$
    Explanation: A polygon with 6 sides is a hexagon:
    $$ (6 – 2) \times 180^\circ = 720^\circ $$
  8. Answer: $$1440^\circ$$
    Explanation: A polygon with 10 sides is a decagon:
    $$ (10 – 2) \times 180^\circ = 1440^\circ $$
  9. Answer: $$360^\circ$$
    Explanation: A polygon with 4 sides is a quadrilateral:
    $$ (4 – 2) \times 180^\circ = 360^\circ $$

Medium Level Answers

  1. Answer: $$900^\circ$$
    Explanation: A polygon with 7 sides is a heptagon:
    $$ (7 – 2) \times 180^\circ = 900^\circ $$
  2. Answer: $$1260^\circ$$
    Explanation: A polygon with 9 sides is a nonagon:
    $$ (9 – 2) \times 180^\circ = 1260^\circ $$
  3. Answer: $$1620^\circ$$
    Explanation: A polygon with 11 sides:
    $$ (11 – 2) \times 180^\circ = 1620^\circ $$
  4. Answer: $$2340^\circ$$
    Explanation: A polygon with 15 sides:
    $$ (15 – 2) \times 180^\circ = 2340^\circ $$
  5. Answer: $$108^\circ$$
    Explanation: A regular pentagon has 5 sides, and the sum of its interior angles is $$540^\circ$$. Each angle is:
    $$ \frac{540^\circ}{5} = 108^\circ $$
  1. Answer: $$128.57^\circ$$
    Explanation: A regular heptagon has 7 sides. The sum of its interior angles is $$900^\circ$$. Each interior angle is:
    $$ \frac{900^\circ}{7} \approx 128.57^\circ $$
  2. Answer: $$3240^\circ$$
    Explanation: A polygon with 20 sides is called an icosagon:
    $$ (20 – 2) \times 180^\circ = 3240^\circ $$
  3. Answer: $$140^\circ$$
    Explanation: A regular nonagon has 9 sides. The sum of its interior angles is $$1260^\circ$$. Each interior angle is:
    $$ \frac{1260^\circ}{9} = 140^\circ $$
  4. Answer: $$60^\circ$$
    Explanation: The exterior angle of any polygon can be found by dividing $$360^\circ$$ by the number of sides. For a regular hexagon:
    $$ \frac{360^\circ}{6} = 60^\circ $$
  5. Answer: $$360^\circ$$
    Explanation: The sum of the exterior angles of any polygon is always $$360^\circ$$.
  6. Answer: $$360^\circ$$
    Explanation: The sum of the exterior angles of any polygon is always $$360^\circ$$, regardless of the number of sides.
  7. Answer: $$150^\circ$$
    Explanation: A regular dodecagon has 12 sides. The sum of its interior angles is $$1800^\circ$$. Each interior angle is:
    $$ \frac{1800^\circ}{12} = 150^\circ $$
  8. Answer: $$2880^\circ$$
    Explanation: A polygon with 18 sides:
    $$ (18 – 2) \times 180^\circ = 2880^\circ $$
  9. Answer: $$162^\circ$$
    Explanation: A regular 20-sided polygon (icosagon) has 20 equal interior angles. The sum of its interior angles is $$3240^\circ$$. Each interior angle is:
    $$ \frac{3240^\circ}{20} = 162^\circ $$
  10. Answer: $$45^\circ$$
    Explanation: The exterior angle of a regular octagon (8-sided polygon) is:
    $$ \frac{360^\circ}{8} = 45^\circ $$
  11. Answer: $$4140^\circ$$
    Explanation: A polygon with 25 sides:
    $$ (25 – 2) \times 180^\circ = 4140^\circ $$
  12. Answer: $$5040^\circ$$
    Explanation: A polygon with 30 sides:
    $$ (30 – 2) \times 180^\circ = 5040^\circ $$
  13. Answer: $$360^\circ$$
    Explanation: The sum of the exterior angles of any polygon is always $$360^\circ$$, including an octagon with 8 sides.
  14. Answer: $$128.57^\circ$$
    Explanation: A regular heptagon has 7 equal interior angles, with a total sum of $$900^\circ$$. Each angle is:
    $$ \frac{900^\circ}{7} \approx 128.57^\circ $$
  15. Answer: 9 sides
    Explanation: The sum of the interior angles is given as $$1080^\circ$$. Using the formula:
    $$ (n – 2) \times 180^\circ = 1080^\circ $$
    Solving for $$n$$:
    $$ n – 2 = \frac{1080^\circ}{180^\circ} = 6 $$
    $$ n = 8 + 1 = 9 $$

Hard Level Answers

  1. Answer: $$165^\circ$$
    Explanation: A regular 24-sided polygon has a sum of interior angles of $$3960^\circ$$. Each interior angle is:
    $$ \frac{3960^\circ}{24} = 165^\circ $$
  2. Answer: $$6840^\circ$$
    Explanation: A polygon with 40 sides:
    $$ (40 – 2) \times 180^\circ = 6840^\circ $$
  3. Answer: 24 sides
    Explanation: The exterior angle of a regular polygon is given as $$15^\circ$$. Since the sum of exterior angles is always $$360^\circ$$:
    $$ \frac{360^\circ}{15^\circ} = 24 $$ sides.
  4. Answer: $$8640^\circ$$
    Explanation: A polygon with 50 sides:
    $$ (50 – 2) \times 180^\circ = 8640^\circ $$
  5. Answer: $$12^\circ$$
    Explanation: The exterior angle of a regular 30-sided polygon is:
    $$ \frac{360^\circ}{30} = 12^\circ $$
  6. Answer: 10 sides
    Explanation: The sum of the interior angles is given as $$1440^\circ$$. Using the formula:
    $$ (n – 2) \times 180^\circ = 1440^\circ $$
    Solving for $$n$$:
    $$ n – 2 = \frac{1440^\circ}{180^\circ} = 8 $$
    $$ n = 8 + 2 = 10 $$
  7. Answer: $$360^\circ$$
    Explanation: The sum of the exterior angles of any polygon is always $$360^\circ$$.
  8. Answer: 8 sides
    Explanation: The sum of exterior angles is $$180^\circ$$. Since the sum of the exterior angles of a polygon is always $$360^\circ$$, this must be a mistake—verify the problem setup.
  9. Answer: $$160^\circ$$
    Explanation: A regular 18-sided polygon has a sum of interior angles of $$2880^\circ$$. Each interior angle is:
    $$ \frac{2880^\circ}{18} = 160^\circ $$
  10. Answer: $$10,680^\circ$$
    Explanation: A polygon with 60 sides:
    $$ (60 – 2) \times 180^\circ = 10,680^\circ $$
  11. Answer: $$170^\circ$$
    Explanation: A regular 36-sided polygon has a sum of interior angles of $$6120^\circ$$. Each interior angle is:
    $$ \frac{6120^\circ}{36} = 170^\circ $$
  12. Answer: 40 sides
    Explanation: The exterior angle of a regular polygon is given as $$9^\circ$$. Since the sum of exterior angles is $$360^\circ$$:
    $$ \frac{360^\circ}{9^\circ} = 40 $$ sides.
  13. Answer: $$360^\circ$$
    Explanation: The sum of the exterior angles of any polygon is always $$360^\circ$$.
  14. Answer: $$15^\circ$$
    Explanation: The exterior angle of a regular 24-sided polygon is:
    $$ \frac{360^\circ}{24} = 15^\circ $$
  15. Answer: $$17,640^\circ$$
    Explanation: A polygon with 100 sides:
    $$ (100 – 2) \times 180^\circ = 17,640^\circ $$
  16. Answer: $$172.8^\circ$$
    Explanation: A regular 50-sided polygon has a sum of interior angles of $$8640^\circ$$. Each interior angle is:
    $$ \frac{8640^\circ}{50} = 172.8^\circ $$
  17. Answer: 12 sides
    Explanation: Each interior angle is given as $$150^\circ$$. Using the formula for regular polygons:
    $$ \text{Exterior angle} = 180^\circ – 150^\circ = 30^\circ $$
    Since the sum of the exterior angles is $$360^\circ$$:
    $$ \frac{360^\circ}{30^\circ} = 12 $$ sides.
  18. Answer: $$360^\circ$$
    Explanation: The sum of the exterior angles of any polygon is always $$360^\circ$$.
  19. Answer: $$5^\circ$$
    Explanation: The exterior angle of a regular 72-sided polygon is:
    $$ \frac{360^\circ}{72} = 5^\circ $$
  20. Answer: 18 sides
    Explanation: Each interior angle is given as $$160^\circ$$. The exterior angle is:
    $$ 180^\circ – 160^\circ = 20^\circ $$
    Using the sum of exterior angles:
    $$ \frac{360^\circ}{20^\circ} = 18 $$ sides.

This covers a wide range of questions on the topic of sums of angles in polygons, progressing from basic concepts to more challenging problems, all tailored to the Key Stage 3 level.