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Shape and Space - Properties and Classification of Polygons

Grade 6IB

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

🔑Concepts

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A polygon is a closed two-dimensional shape formed by three or more straight line segments. A polygon with nn sides is called an nn-gon.

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Regular Polygons have all sides of equal length and all interior angles of equal measure (e.g., an equilateral triangle or a square).

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Irregular Polygons do not have all sides and angles equal.

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A Convex Polygon has no interior angles greater than 180∘180^\circ. A Concave Polygon has at least one interior angle greater than 180∘180^\circ (a reflex angle).

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Triangles are classified by sides into Equilateral (33 equal sides), Isosceles (22 equal sides), and Scalene (00 equal sides).

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Triangles are classified by angles into Acute (all angles <90∘< 90^\circ), Right (one angle =90∘= 90^\circ), and Obtuse (one angle >90∘> 90^\circ).

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Quadrilaterals are four-sided polygons. Common types include Parallelograms (opposite sides parallel), Rectangles (four 90∘90^\circ angles), Rhombuses (four equal sides), and Trapeziums (one pair of parallel sides).

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The sum of the interior angles of a triangle is always 180∘180^\circ.

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The sum of the interior angles of a quadrilateral is always 360∘360^\circ.

📐Formulae

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

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

Sum of exterior angles=360∘\text{Sum of exterior angles} = 360^\circ

Exterior angle of a regular polygon=360∘n\text{Exterior angle of a regular polygon} = \frac{360^\circ}{n}

💡Examples

Problem 1:

Calculate the sum of the interior angles of a hexagon (n=6n = 6).

Solution:

Sum=(6−2)×180∘=4×180∘=720∘Sum = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ

Explanation:

Using the formula for the sum of interior angles where nn is the number of sides, we substitute n=6n = 6 for a hexagon.

Problem 2:

Find the size of one interior angle of a regular octagon.

Solution:

Interior Angle=(8−2)×180∘8=6×180∘8=1080∘8=135∘\text{Interior Angle} = \frac{(8 - 2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ

Explanation:

A regular octagon has 88 equal sides and 88 equal angles. We use the formula for a single interior angle with n=8n = 8.

Problem 3:

Three angles of a quadrilateral are 95∘95^\circ, 85∘85^\circ, and 70∘70^\circ. Find the measure of the fourth angle.

Solution:

Fourth Angle=360∘−(95∘+85∘+70∘)=360∘−250∘=110∘\text{Fourth Angle} = 360^\circ - (95^\circ + 85^\circ + 70^\circ) = 360^\circ - 250^\circ = 110^\circ

Explanation:

The sum of interior angles of any quadrilateral is 360∘360^\circ. Subtracting the sum of the three known angles from 360∘360^\circ gives the missing angle.

Properties and Classification of Polygons Grade 6 Notes & Examples