2D Shapes
Section: Geometry | Syllabus: Cambridge Primary Mathematics (0845)
Quadrilaterals
A quadrilateral is any 2D shape with 4 straight sides. Different quadrilaterals are classified by looking at their angles, whether their sides are parallel, and how their diagonals behave.
- A quadrilateral always has 4 sides and 4 angles, and its angles always add up to 360°
- Sides are parallel when they point in exactly the same direction and never meet, however far they are extended
- A diagonal is a straight line joining two corners of a shape that are not already joined by a side
Each quadrilateral is classified by its sides, angles and diagonals
| Shape | Sides | Angles | Diagonals |
|---|---|---|---|
| Square | 4 equal, 2 pairs parallel | 4 right angles | Equal length, cross at right angles |
| Rectangle | 2 pairs equal, 2 pairs parallel | 4 right angles | Equal length, do not cross at right angles |
| Parallelogram | 2 pairs equal, 2 pairs parallel | Opposite angles equal | Unequal length, bisect each other |
| Rhombus | 4 equal, 2 pairs parallel | Opposite angles equal | Unequal length, cross at right angles |
| Trapezium | At least 1 pair parallel | Varies | Unequal length |
| Kite | 2 pairs of adjacent equal sides | 1 pair opposite angles equal | Cross at right angles, only 1 is bisected |
Worked Example: Identifying a Quadrilateral from Its Properties
- Question: A quadrilateral has 4 equal sides, and its diagonals cross at right angles but are not equal in length. Which quadrilateral is it?
- Step 1: 4 equal sides with diagonals crossing at right angles rules out a rectangle and a parallelogram - both have unequal sides
- Step 2: A square also has 4 equal sides and right-angle diagonals, but a square's diagonals are equal in length
- Step 3: A shape with 4 equal sides, right-angle diagonals, but unequal diagonal lengths cannot have right-angle corners
- Answer: Rhombus
Common Mistakes
MistakeThinking a rhombus is always a square
Fixevery square is a rhombus, but a rhombus only needs 4 equal sides - it does not need right angles
MistakeThinking a trapezium has no parallel sides
Fixa trapezium has at least one pair of parallel sides - that is exactly what makes it a trapezium
Parts of a Circle
Every circle is described using the same four key parts, and knowing how they relate to each other makes many circle calculations straightforward.
- The centre is the single point in the middle of the circle, the same distance from every point on the edge
- The radius is the distance from the centre to any point on the edge
- The diameter is the distance straight across the circle through the centre - it is always twice the radius
- The circumference is the distance all the way around the edge of the circle
Diameter = 2 × radius, since the diameter passes through the centre from one side to the other
Worked Example: Finding the Diameter from the Radius
- Question: A circle has a radius of 6 cm. What is its diameter?
- Step 1: The diameter is always twice the radius
- Step 2: 6 × 2 = 12
- Answer: 12 cm
Common Mistakes
MistakeConfusing the radius and the diameter
Fixthe radius goes from the centre to the edge; the diameter goes all the way across through the centre, so it is always double the radius
Constructing Circles
A circle of a given size can be drawn accurately using a compass, by setting the gap between the point and the pencil to match the radius.
- To construct a circle of a given radius, open the compass so the gap between the point and the pencil tip matches that radius, place the point at the centre, then turn the compass through a full circle
- To construct a circle of a given diameter, first halve the diameter to find the radius, then set the compass to that radius
The compass point stays fixed at the centre while the pencil traces a path that is always the same distance away - the radius
Worked Example: Constructing a Circle from a Diameter
- Question: Construct a circle with a diameter of 10 cm.
- Step 1: Halve the diameter to find the radius: 10 ÷ 2 = 5 cm
- Step 2: Set the compass to a radius of 5 cm
- Step 3: Place the compass point at the centre and turn it through a full circle
- Answer: A circle of diameter 10 cm, drawn using a radius of 5 cm
Common Mistakes
MistakeSetting the compass to the diameter instead of the radius when only a diameter is given
Fixa compass is always set to the radius - halve the diameter first if that is what you are given
Rotational Symmetry
A shape has rotational symmetry if it looks exactly the same after being turned part-way around its centre, before completing a full turn. The order of rotational symmetry counts how many times this happens in one full 360° turn.
- The order of rotational symmetry is the number of positions in one full turn where the shape looks identical to its start
- Every shape has an order of at least 1, since it always matches itself after a complete 360° turn
- A shape is described as having "order x", where x is that number of matching positions
A square matches its original position 4 times in one full turn - at 90°, 180°, 270° and 360° - so it has order 4
Worked Example: Finding the Order of Rotational Symmetry
- Question: What is the order of rotational symmetry of a rectangle that is not a square?
- Step 1: Turn the rectangle 90° - it now looks different, since the long and short sides have swapped
- Step 2: Turn it to 180° - it looks identical to the start
- Step 3: Turn it to 270° - it looks different again
- Step 4: Turn it back to 360° - it matches the start, as every shape does
- Answer: The rectangle matches at 180° and 360° - order 2
Common Mistakes
MistakeConfusing rotational symmetry with line symmetry
Fixline symmetry is about folding a shape onto itself; rotational symmetry is about turning it - a shape can have one without the other
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