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Geometry - Properties of 2D shapes (triangles, quadrilaterals, polygons)

Grade 5IGCSE

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

🔑Concepts

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Classification of Triangles: By sides (Equilateral, Isosceles, Scalene) and by angles (Acute, Right, Obtuse).

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Properties of Triangles: The sum of interior angles is always 180°.

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Quadrilateral Types: Identifying Square, Rectangle, Parallelogram, Rhombus, Trapezium, and Kite based on side lengths, parallel lines, and angles.

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Properties of Quadrilaterals: The sum of interior angles is always 360°.

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Regular vs. Irregular Polygons: Regular polygons have all sides and all interior angles equal.

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Polygon Naming: Pentagon (5 sides), Hexagon (6 sides), Octagon (8 sides).

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Symmetry: Line symmetry (mirror lines) and Rotational symmetry (how many times a shape looks the same in a full turn).

📐Formulae

Sum of interior angles in any triangle = 180∘180^{\circ}

Sum of interior angles in any quadrilateral = 360∘360^{\circ}

Sum of interior angles in an nn-sided polygon = (n−2)×180∘(n - 2) \times 180^{\circ}

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

💡Examples

Problem 1:

An isosceles triangle has one angle of 70∘70^{\circ} at its apex (the angle between the two equal sides). Calculate the size of the other two angles.

Solution:

55∘55^{\circ} each

Explanation:

Since the triangle is isosceles, the two base angles are equal. The sum of angles is 180∘180^{\circ}. Subtract the apex angle: 180∘−70∘=110∘180^{\circ} - 70^{\circ} = 110^{\circ}. Divide by 2 for the two equal angles: 110∘÷2=55∘110^{\circ} \div 2 = 55^{\circ}.

Problem 2:

A quadrilateral has three angles measuring 90∘90^{\circ}, 85∘85^{\circ}, and 105∘105^{\circ}. Find the value of the fourth angle.

Solution:

80∘80^{\circ}

Explanation:

The sum of angles in a quadrilateral is 360∘360^{\circ}. Add the known angles: 90+85+105=280∘90 + 85 + 105 = 280^{\circ}. Subtract from the total: 360∘−280∘=80∘360^{\circ} - 280^{\circ} = 80^{\circ}.

Problem 3:

What is the sum of the interior angles of a regular Hexagon?

Solution:

720∘720^{\circ}

Explanation:

A hexagon has n=6n = 6 sides. Using the formula (n−2)×180∘(n - 2) \times 180^{\circ}, we get (6−2)×180∘=4×180∘=720∘(6 - 2) \times 180^{\circ} = 4 \times 180^{\circ} = 720^{\circ}.