Geometry is the study of shapes, sizes, and the properties of space. In this chapter, we explore key geometric ideas: properties of quadrilaterals, polygons, symmetry, and three-dimensional figures. Understanding these themes builds a foundation for higher geometry.
Polygons
A polygon is a closed figure made of three or more line segments. Polygons are classified by the number of sides:
Triangle (3), Quadrilateral (4), Pentagon (5), Hexagon (6), Heptagon (7), Octagon (8).
Sum of interior angles of an n-sided polygon = (n - 2) × 180 degrees.
Sum of exterior angles of any convex polygon = 360 degrees.
Regular polygon: all sides equal and all angles equal.
Interior angle of a regular n-gon = [(n - 2) × 180] / n.
Quadrilaterals and Their Properties
A parallelogram has opposite sides parallel and equal, opposite angles equal, diagonals bisect each other.
A rectangle is a parallelogram with all right angles; diagonals are equal.
A rhombus is a parallelogram with all sides equal; diagonals are perpendicular bisectors of each other.
A square is both a rectangle and a rhombus.
A trapezium has exactly one pair of parallel sides.
A kite has two pairs of adjacent sides equal; one diagonal bisects the other at right angles.
Find the sum of interior angles of a heptagon (7 sides).
Sum = (7 - 2) × 180 = 5 × 180 = 900 degrees.
Each interior angle of a regular polygon is 135 degrees. How many sides?
(n-2) × 180 / n = 135 → (n-2) × 180 = 135n → 180n - 360 = 135n → 45n = 360 → n = 8 sides (octagon).
In a parallelogram, one angle is 70 degrees. Find all four angles.
Adjacent angles are supplementary: 70 + x = 180 → x = 110.
Angles: 70, 110, 70, 110 degrees.
The diagonals of a rhombus are 16 cm and 12 cm. Find the side.
Each half-diagonal: 8 cm and 6 cm. Side = √(82 + 62) = √100 = 10 cm.
A rectangle has length 12 cm and width 5 cm. Find its diagonal.
Diagonal = √(122 + 52) = √(144 + 25) = √169 = 13 cm.
Find each exterior angle of a regular hexagon.
Sum of exterior angles = 360. Each = 360/6 = 60 degrees.
In a kite ABCD, AB = AD = 5 cm and CB = CD = 8 cm. Diagonal AC = 10 cm. Find the area.
Area of kite = (d1 × d2)/2, where d1 and d2 are diagonals. Need d2. In the right triangles: d2/2 = √(52 - ?) — use the property: diagonal AC bisects at right angles. Area = (d1 × d2)/2 once both diagonals are known. Formula: area of kite = product of diagonals / 2.
- Key formulas:
- Interior angle sum of n-gon: (n-2) × 180
- Each interior angle of regular n-gon: (n-2) × 180 / n
- Each exterior angle of regular n-gon: 360/n
- Area of rhombus/kite: (d1 × d2)/2
Common mistakes
- Using the formula (n-2)×180 for exterior angles instead of interior angles.
- Confusing properties of a rhombus (perpendicular diagonals) with a rectangle (equal diagonals).
- Thinking all quadrilaterals with equal sides are squares.
Summary
Each type of polygon has unique angle and side properties. The interior angle sum formula is key. Quadrilaterals form a rich family — rectangle, rhombus, square, trapezium, kite — each with its own special properties worth memorising.