Geometry Toolkit

Rotational Symmetry

• **Rotational symmetry** is the number of times a shape looks the same when rotated 360° about its centre.
• This number is called the **order of rotational symmetry**.
• Use tracing paper with an arrow to track orientation; returning to start counts as 1.
• A shape with order 1 is said to have **no rotational symmetry**.
• Example: a square has rotational symmetry of order 4.

Lines of Symmetry

• A **line of symmetry** divides a shape into two mirror-image halves.
• Folding along a line of symmetry makes the two parts coincide exactly.
• Some shapes have multiple lines (e.g., square has 4, rectangle has 2).
• For diagonal lines, use tracing paper to reflect the shape.
• When completing a shape given a line of symmetry, reflect the given part across the line.

2D Shapes

• Polygons are named by number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10).
• A **regular polygon** has all sides equal and all angles equal.
• Triangles: **equilateral** (3 equal sides), **isosceles** (2 equal sides), **right-angled** (90° angle), **scalene** (no equal sides).
• Quadrilaterals: **square** (4 equal sides, 4 right angles, 4 lines of symmetry, order 4 rotational symmetry).
• **Rectangle**: opposite sides equal, 2 lines of symmetry, order 2 rotational symmetry.
• **Parallelogram**: opposite sides parallel and equal, no lines of symmetry, order 2 rotational symmetry.
• **Rhombus**: all sides equal, opposite angles equal, 2 lines of symmetry, order 2 rotational symmetry.
• **Trapezium**: one pair of parallel sides; isosceles trapezium has non-parallel sides equal and 1 line of symmetry.
• **Kite**: two pairs of equal adjacent sides, 1 line of symmetry, order 1 rotational symmetry.
• Circle terms: **circumference** (perimeter), **diameter** (line through centre), **radius** (centre to edge), **arc**, **sector**, **chord**, **segment**, **tangent**.

3D Shapes

• Common 3D shapes: **cube**, **cuboid**, **cylinder**, **prism**, **pyramid**, **cone**, **sphere**, **tetrahedron**.
• A **prism** has the same cross-section throughout; a **cylinder** is like a prism with circular cross-section.
• A **pyramid** has a flat base and sloping sides meeting at a point (apex).
• Faces, vertices, edges: cube (6 faces, 8 vertices, 12 edges); cuboid (6 faces, 8 vertices, 12 edges); tetrahedron (4 faces, 4 vertices, 6 edges).
• A **net** is a 2D drawing that folds to form a 3D shape; the net of a cube has 6 squares (11 possible arrangements).
• Net of a cylinder: two circles and a rectangle (rectangle length = circumference of circle).
• Net of a pyramid: base plus a triangle attached to each edge of the base.

Planes of Symmetry

• A **plane of symmetry** splits a 3D shape into two congruent mirror-image halves.
• Cubes have **9** planes of symmetry; cuboids have **3** (if all dimensions different).
• Cylinders have an infinite number of planes of symmetry (any plane through the axis).
• For a prism, number of planes = number of lines of symmetry in $cross-section + 1$.
• For a pyramid with a regular n-sided base, number of planes $= n (the$ lines of symmetry of the base).

Converting between Units

• Length: 1 $cm = 10$ mm, $1 m = 100$ cm, 1 $km = 1000 m$.
• Mass: $1 g = 1000$ mg, 1 $kg = 1000 g, 1$ tonne $= 1000$ kg.
• Capacity: 1 litre $= 100$ $cl = 1000$ ml; 1 $ml = 1$ cm³; 1 litre $= 1000$ cm³; $1 m^{3} = 1000$ litres.
• To convert, multiply or divide by the conversion factor; check if the number of units increases or decreases.

Squared & Cubic Units

• For area (squared units), square the linear conversion factor: $1 cm^{2} = 100 mm^{2}, 1 m^{2} = 10 000 cm^{2}, 1 km^{2} = 1 000 000 m^{2}$.
• 1 hectare $= 10 000 m^{2}$.
• For volume (cubic units), cube the linear conversion factor: $1 cm^{3} = 1000 mm^{3}, 1 m^{3} = 1 000 000 cm^{3}, 1 km^{3} = 1 000 000 000 m^{3}$.
• Example: $2.54 m^{3} = 2.54 \times 1 000 000 = 2 540 000 cm^{3}$.

Rotational Symmetry of a Square

Lines of Symmetry of a Square

Planes of Symmetry of a Cuboid

Net of a Cube