krit.club logo

Geometry - Polygons and Symmetry

Grade 9IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

A polygon is a closed 2D shape with straight sides. A regular polygon has all sides and all interior angles equal. The exterior angle is formed by extending one side of the polygon. At any vertex, the interior angle and the exterior angle sum to 180∘180^\circ because they lie on a straight line.

Diagram showing the relationship between interior and exterior angles on a polygon vertex.
•

The sum of the exterior angles of any convex polygon is always 360∘360^\circ. For a regular polygon with nn sides, each exterior angle is 360∘n\frac{360^\circ}{n}. This property is useful for finding the number of sides when the angle is known.

•

The sum of interior angles of a polygon with nn sides is (n−2)×180∘(n - 2) \times 180^\circ. This is derived by dividing the polygon into (n−2)(n-2) triangles from a single vertex, where each triangle contributes 180∘180^\circ.

A pentagon divided into three triangles to demonstrate the interior angle sum formula.
•

Symmetry in polygons includes line symmetry (reflectional) and rotational symmetry. A regular polygon with nn sides has nn lines of symmetry and rotational symmetry of order nn.

📐Formulae

Sum of interior angles = (n−2)×180∘(n - 2) \times 180^\circ

Individual interior angle of a regular polygon = (n−2)×180∘n\frac{(n - 2) \times 180^\circ}{n}

Sum of exterior angles = 360∘360^\circ

Individual exterior angle of a regular polygon = 360∘n\frac{360^\circ}{n}

Interior angle + Exterior angle = 180∘180^\circ

💡Examples

Problem 1:

Calculate the size of each interior angle of a regular decagon (10-sided polygon).

Solution:

144°

Explanation:

Using the formula for a regular polygon: ((10−2)×180)/10=(8×180)/10=1440/10=144∘((10-2) \times 180) / 10 = (8 \times 180) / 10 = 1440 / 10 = 144^\circ. Alternatively, find the exterior angle first: 360/10=36∘360 / 10 = 36^\circ, then subtract from 180: 180−36=144∘180 - 36 = 144^\circ.

Problem 2:

A regular polygon has an exterior angle of 24°. How many sides does it have?

Solution:

15

Explanation:

The sum of exterior angles is always 360°. To find the number of sides nn, use the formula n=360/exterior anglen = 360 / \text{exterior angle}. Therefore, n=360/24=15n = 360 / 24 = 15 sides.

Problem 3:

Describe the symmetry of a rhombus.

Solution:

Line symmetry: 2; Rotational symmetry: Order 2

Explanation:

A rhombus has two lines of symmetry (the diagonals). It also has rotational symmetry of order 2 because it looks the same twice (at 180° and 360°) during a full rotation.

Problem 4:

The diagram shows a regular hexagon. Calculate the value of the interior angle xx.

Regular hexagon with an interior angle labeled x.

Solution:

  1. A hexagon has n=6n = 6 sides.
  2. Use the formula for the sum of interior angles: (6−2)×180∘=4×180∘=720∘(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
  3. Since it is a regular hexagon, all 6 interior angles are equal.
  4. x=720∘6=120∘x = \frac{720^\circ}{6} = 120^\circ.

Explanation:

To find the interior angle of a regular polygon, divide the total sum of angles by the number of vertices.

Problem 5:

An irregular pentagon has four interior angles of 90∘90^\circ, 110∘110^\circ, 120∘120^\circ, and 130∘130^\circ. Find the size of the fifth interior angle yy.

Irregular pentagon with four interior angles and one unknown angle y.

Solution:

  1. A pentagon has n=5n = 5 sides.
  2. Sum of interior angles = (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
  3. Sum of the known angles = 90∘+110∘+120∘+130∘=450∘90^\circ + 110^\circ + 120^\circ + 130^\circ = 450^\circ.
  4. y=540∘−450∘=90∘y = 540^\circ - 450^\circ = 90^\circ.

Explanation:

For any polygon, the sum of interior angles is constant based on the number of sides. Subtracting the known angles from this sum gives the missing value.