Plane Shapes and Polygons

Properties of triangles, quadrilaterals, exterior angle theorem, and polygon interior angle sum formula.

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Curated Video Lesson

Visual explanation and practical step-by-step walk-through

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Comprehensive Study Notes

Aligned with Ghana NaCCA & WAEC BECE syllabus standards

Topic Introduction & Real-World Context:

A polygon is a closed two-dimensional plane figure made of straight line segments. From triangular roof trusses to rectangular building plots and hexagonal tiling patterns, polygons govern architecture. In this topic, students explore the angle properties of triangles, special quadrilaterals, and regular polygons.

What You Will Master in This Lesson (NaCCA Objectives):

Classify triangles by sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse).

Apply the triangle angle sum theorem (180°) and exterior angle theorem.

Identify properties of quadrilaterals: squares, rectangles, parallelograms, rhombuses, trapeziums, and kites.

Calculate the sum of interior angles of any n-sided polygon using (n - 2) × 180°.

Calculate individual interior and exterior angles of regular polygons.

1. Angle Properties of Triangles

• Sum of interior angles of ANY triangle = 180°. • Exterior Angle Theorem: The exterior angle of a triangle equals the sum of the two opposite interior angles. • Equilateral: All 3 sides equal, all 3 angles = 60°. • Isosceles: 2 sides equal, 2 base angles equal. • Scalene: All 3 sides and angles are different.

2. Quadrilaterals and Their Properties

Sum of interior angles of any quadrilateral = 360°. • Square: 4 equal sides, 4 right angles, diagonals equal and bisect at 90°. • Rectangle: Opposite sides equal, 4 right angles, diagonals equal. • Parallelogram: Opposite sides parallel and equal, opposite angles equal, diagonals bisect each other. • Rhombus: Parallelogram with 4 equal sides, diagonals bisect at 90°. • Trapezium: Exactly one pair of parallel sides.

3. Polygon Interior and Exterior Angles

For any n-sided polygon: • Sum of Interior Angles: S = (n - 2) × 180°. - Quadrilateral (n = 4): (4 - 2) × 180° = 360°. - Pentagon (n = 5): (5 - 2) × 180° = 540°. - Hexagon (n = 6): (6 - 2) × 180° = 720°. - Octagon (n = 8): (8 - 2) × 180° = 1080°. • Sum of Exterior Angles of ANY convex polygon = 360°. • For a Regular Polygon (all sides and angles equal): - Each Exterior Angle = 360° / n. - Each Interior Angle = 180° - (Each Exterior Angle).

Common Mistakes Students Make in BECE Examinations:

⚠️Using (n - 2) × 360° instead of (n - 2) × 180° for interior angle sum.

⚠️Confusing the exterior angle with reflex angles.

⚠️Forgetting that base angles of an isosceles triangle are opposite the equal sides.

Teacher's BECE Exam Pro-Tips:

⭐To find the interior angle of a regular polygon easily: calculate the exterior angle first (360° / n), then subtract from 180°!

⭐Sum of exterior angles is ALWAYS 360°, regardless of the number of sides.

Quick Revision Summary Checklist:

Triangle angle sum = 180°. Exterior angle = sum of opposite interior angles.

Polygon interior sum = (n - 2) × 180°.

Polygon exterior sum = 360°.

Regular polygon: Exterior = 360° / n; Interior = 180° - (360° / n).

Step-by-Step Worked Examples (1)

Real BECE exam-standard problems with complete solution steps

Example 1: Interior Angle Sum of a Polygon

Problem StatementCalculate the sum of interior angles of a hexagon (6-sided polygon).

Step-by-Step Solution:

Step 1: Formula: S = (n - 2) × 180° where n = 6.

Step 2: S = (6 - 2) × 180° = 4 × 180° = 720°.

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Key Takeaway / Exam Rule: The sum of interior angles increases by 180° for each additional side.

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