Geometry

Angles

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

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:

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°.

Perimeter

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

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).

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

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:

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.