Geometry

Angles

An angle is formed where two rays meet at a point (the vertex). Angles are measured in degrees (°).

Angle type	Size
Acute	Less than 90°
Right	Exactly 90°
Obtuse	Between 90° and 180°
Straight	Exactly 180°
Reflex	Greater than 180°
Full turn	Exactly 360°

Angle Rules

These facts appear constantly in geometry problems:

• Angles on a straight line sum to 180°
• Angles around a point sum to 360°
• Vertically opposite angles (formed when two lines cross) are equal
• Angles in a triangle sum to 180°
• Angles in a quadrilateral sum to 360°

Example: In a triangle, two angles are 65° and 78°. Find the third. 180 - 65 - 78 = 37°

Types of Triangles

By side length:

Name	Sides	Angles
Equilateral	All 3 equal	All 60°
Isosceles	2 sides equal	2 angles equal
Scalene	No sides equal	No angles equal

By angles:

• Right-angled — one angle is exactly 90°
• Acute — all angles are less than 90°
• Obtuse — one angle is greater than 90°

A triangle can be both isosceles and right-angled (e.g., 90°, 45°, 45°).

Quadrilaterals

A quadrilateral is any four-sided polygon. Angles in a quadrilateral always sum to 360°.

Shape	Key properties
Square	4 equal sides, 4 right angles, all properties of rectangle and rhombus
Rectangle	Opposite sides equal, 4 right angles, diagonals equal
Rhombus	4 equal sides, opposite angles equal, diagonals bisect at right angles
Parallelogram	Opposite sides parallel and equal, opposite angles equal, diagonals bisect each other
Trapezium	Exactly one pair of parallel sides
Kite	Two pairs of adjacent equal sides, one pair of equal angles

Perimeter

Perimeter is the total distance around the outside of a shape. Add up all the side lengths.

Shape	Formula
Rectangle	P = 2(l + w)
Square	P = 4s
Triangle	P = a + b + c
Regular polygon	P = n × s (n sides, each length s)

Example: Rectangle with l = 8, w = 3: P = 2(8 + 3) = 2 × 11 = 22

Area

Area is the amount of space inside a 2D shape, measured in square units (cm², m²).

Shape	Formula
Rectangle	A = l × w
Square	A = s²
Triangle	A = ½ × base × height
Parallelogram	A = base × height
Trapezium	A = ½ × (a + b) × h
Circle	A = π × r²

Example — Triangle: base = 10, height = 6: A = ½ × 10 × 6 = 30

Example — Rectangle: l = 9, w = 4: A = 9 × 4 = 36

Note: for a triangle, the height must be perpendicular (at right angles) to the base, not along a slanted side.

Circles

Term	Meaning	Formula
Radius (r)	Distance from centre to edge	—
Diameter (d)	Distance across circle	d = 2r
Circumference	Perimeter of circle	C = 2πr = πd
Area	Space inside circle	A = πr²

π (pi) ≈ 3.14159... Use 3.14 for approximations, or leave answers in terms of π for exactness.

Example: Circle with radius 5:

• Circumference = 2 × 3.14 × 5 = 31.4
• Area = 3.14 × 5² = 3.14 × 25 = 78.5

Example: Circle with radius 4:

• Area = 3.14 × 16 = 50.24

3D Shapes

Common 3D solids and their properties:

Shape	Faces	Edges	Vertices
Cube	6	12	8
Cuboid	6	12	8
Triangular prism	5	9	6
Square pyramid	5	8	5
Cylinder	3	2	0
Sphere	1	0	0

Euler's formula for polyhedra: Faces + Vertices - Edges = 2

Common Mistakes

• Using the slant height instead of perpendicular height for triangle and parallelogram area.
• Confusing perimeter and area — perimeter is a length (cm), area is a space (cm²).
• Using diameter instead of radius in circle formulas — always halve the diameter first.
• Forgetting that quadrilateral angles sum to 360°, not 180°.

Tips and Tricks

• For composite shapes (e.g., an L-shape), split into rectangles, find each area, then add (or subtract).
• To find a missing angle in a triangle: subtract the two known angles from 180°.
• Memorise the circle formulas as a pair: C = 2πr (circumference) and A = πr² (area).
• The area of a parallelogram = base × perpendicular height (not the slant side).
• When dealing with circles, check whether the question gives radius or diameter — mistakes here are very common.