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Geometry - Polygons and Symmetry

Grade 9IGCSE

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

🔑Concepts

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Definition of Polygons: A closed 2D shape with straight sides. Regular polygons have all sides and angles equal.

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Interior and Exterior Angles: The sum of exterior angles of any convex polygon is always 360°.

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Properties of Quadrilaterals: Understanding the specific attributes of squares, rectangles, parallelograms, rhombuses, trapeziums, and kites.

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Line Symmetry: A line that divides a shape into two congruent halves that are mirror images of each other.

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Rotational Symmetry: The number of times a shape looks exactly the same during a full 360° rotation (Order of Symmetry).

📐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.