Properties of 2D and 3D Shapes

Knowing the exact properties of shapes is the quiet workhorse of OCR GCSE Mathematics (J560). It is rarely the headline of a question, but it is constantly needed: to justify why an angle is what it is, to identify a shape from a description, or to interpret a plan and elevation. This lesson is a precise reference for the properties of triangles, quadrilaterals and polygons — their sides, angles, diagonals and symmetry — and then moves into three dimensions with solids, nets, and plans and elevations. Recalling a property is AO1; using a property to justify a conclusion is AO2; and applying shape knowledge inside a problem such as a net or an unfamiliar solid is AO3.

Key Vocabulary

Term	Meaning
Diagonal	A line joining two non-adjacent vertices of a polygon
Line of symmetry	A line dividing a shape into two mirror-image halves
Rotational symmetry	The shape looks the same after a turn of less than $360^{\circ}$360∘; the order counts how many times
Perpendicular	At right angles ($90^{\circ}$90∘) to each other
Bisect	To cut exactly in half
Vertex	A corner where edges meet
Net	A 2D pattern that folds up to make a 3D solid
Face / Edge	A flat surface of a solid / a line where two faces meet

Triangles and Their Symmetry

A triangle is classified by its sides and angles, and the two classifications are linked: equal sides force equal angles opposite them. An equilateral triangle has all three sides equal and all three angles $60^{\circ}$60∘, giving it $3$3 lines of symmetry and rotational symmetry of order $3$3. An isosceles triangle has two equal sides and two equal base angles, with exactly $1$1 line of symmetry. A scalene triangle has all sides and all angles different, with no symmetry at all. A right-angled triangle simply has one $90^{\circ}$90∘ angle. Spotting that two sides are equal — perhaps because they are radii of the same circle, or sides of a regular polygon — is one of the most useful moves in the whole geometry strand.

Triangle	Sides	Angles	Lines of symmetry	Rotational order
Equilateral	all $3$3 equal	all $60^{\circ}$60∘	$3$3	$3$3
Isosceles	$2$2 equal	$2$2 equal	$1$1	$1$1
Scalene	all different	all different	$0$0	$1$1
Right-angled	varies	one $90^{\circ}$90∘	$0$0 (or $1$1 if isosceles)	$1$1

Worked Example 1

Triangle $ABC$ABC has $AB = AC = 7$AB=AC=7 cm and $\angle BAC = 50^{\circ}$∠BAC=50∘. Name the type of triangle and work out $\angle ABC$∠ABC.

Since $AB = AC$AB=AC, the triangle is isosceles, so the base angles $\angle ABC$∠ABC and $\angle ACB$∠ACB are equal. Then $50^{\circ} + 2 \times \angle ABC = 180^{\circ}$50∘+2×∠ABC=180∘, so $\angle ABC = 65^{\circ}$∠ABC=65∘.

Answer: isosceles; $\angle ABC = 65^{\circ}$∠ABC=65∘.

Worked Example 1b

A triangle has angles $40^{\circ}$40∘, $70^{\circ}$70∘ and $70^{\circ}$70∘. Classify it as fully as you can and state its symmetry.

The angles sum to $180^{\circ}$180∘ as required. Two angles are equal ($70^{\circ}$70∘), so the sides opposite them are equal and the triangle is isosceles; no angle reaches $90^{\circ}$90∘, so it is also acute. An isosceles triangle has exactly $1$1 line of symmetry and rotational symmetry of order $1$1.

Answer: an acute-angled isosceles triangle with $1$1 line of symmetry.

Common error: describing it only as "acute" and missing the isosceles classification that the two equal angles guarantee.

Quadrilaterals and Their Properties

The quadrilaterals are the part of this topic that earn the most marks, because their diagonal properties are so often the deciding clue. The figure shows the family at a glance; the key is to learn not just the sides and angles but exactly what the diagonals do.

Quadrilateral	Sides	Diagonals	Lines of symmetry	Rotational order
Square	all equal	equal, bisect at $90^{\circ}$90∘	$4$4	$4$4
Rectangle	opposite pairs equal	equal, bisect (not at $90^{\circ}$90∘)	$2$2	$2$2
Parallelogram	opposite pairs equal	bisect (not equal, not at $90^{\circ}$90∘)	$0$0	$2$2
Rhombus	all equal	bisect at $90^{\circ}$90∘ (not equal)	$2$2	$2$2
Trapezium	one pair parallel	not equal in general	$0$0 (isosceles: $1$1)	$1$1
Kite	two pairs adjacent equal	one bisects the other at $90^{\circ}$90∘	$1$1	$1$1

It helps to see how the family fits together: a square is a special rectangle (equal sides) and a special rhombus (right angles); a rectangle and a rhombus are both special parallelograms; and a parallelogram is a special trapezium.

Symmetry is part of the description too. A line of symmetry is a fold line that maps a shape exactly onto itself, and the order of rotational symmetry counts how many positions in a full turn look identical to the start. A square scores highly on both ($4$4 lines, order $4$4) because it is so regular; a parallelogram has no lines of symmetry at all yet still has rotational symmetry of order $2$2, because turning it half a revolution about its centre returns it to its original outline. Being able to state both kinds of symmetry for each quadrilateral is a frequent source of quick marks.

Worked Example 2

A quadrilateral has all four sides equal and diagonals that bisect each other at right angles, but its angles are not all $90^{\circ}$90∘. Name the shape.

All sides equal rules in a rhombus or a square; angles not all $90^{\circ}$90∘ rules out the square. The shape is a rhombus.

Answer: rhombus.

Worked Example 3

$PQRS$PQRS is a parallelogram with $\angle P = 65^{\circ}$∠P=65∘. Work out the other three angles, giving reasons.

In a parallelogram, opposite angles are equal and co-interior angles sum to $180^{\circ}$180∘. So $\angle R = 65^{\circ}$∠R=65∘ (opposite to $\angle P$∠P), $\angle Q = 180^{\circ} - 65^{\circ} = 115^{\circ}$∠Q=180∘−65∘=115∘ (co-interior with $\angle P$∠P), and $\angle S = 115^{\circ}$∠S=115∘ (opposite to $\angle Q$∠Q).

Answer: $\angle R = 65^{\circ}$∠R=65∘, $\angle Q = \angle S = 115^{\circ}$∠Q=∠S=115∘.

Worked Example 4

A quadrilateral has diagonals that are equal in length and bisect each other, but are not perpendicular. Identify it.

Equal, bisecting diagonals point to a rectangle or square; "not perpendicular" excludes the square. The shape is a rectangle.

Answer: rectangle.

Common error: confusing a rhombus and a rectangle — a rhombus has perpendicular diagonals of unequal length, a rectangle has equal diagonals that are not perpendicular.

Worked Example 5

A rhombus $ABCD$ABCD has side $13$13 cm and one diagonal $10$10 cm. Work out the length of the other diagonal.

The diagonals bisect each other at right angles, so each half of the known diagonal is $5$5 cm, forming a right-angled triangle with hypotenuse $13$13 cm. By Pythagoras, the other half-diagonal $= \sqrt{13^{2} - 5^{2}} = \sqrt{144} = 12$=132−52​=144​=12 cm, so the full diagonal is $2 \times 12 = 24$2×12=24 cm.

Answer: $24$24 cm.

3D Solids, Faces, Edges and Vertices

A 3D solid is described by its faces (flat surfaces), edges (where two faces meet) and vertices (corners). The single most useful idea here is the prism: a solid with the same cross-section all the way through, so a slice taken anywhere along its length is identical. A cylinder is just a prism with a circular cross-section. A pyramid, by contrast, narrows to a single point called the apex, and a cone is a pyramid with a circular base. A sphere is perfectly round, with every point on its surface the same distance from the centre, so it has no flat faces, no straight edges and no vertices at all. You should be able to count faces, edges and vertices for the common solids and, just as importantly, recognise each solid from a written description of its faces.

Solid	Faces	Edges	Vertices
Cube	$6$6	$12$12	$8$8
Cuboid	$6$6	$12$12	$8$8
Triangular prism	$5$5	$9$9	$6$6
Square-based pyramid	$5$5	$8$8	$5$5
Cylinder	$3$3 (two circles + curved)	$2$2	$0$0

Worked Example 6

A solid has $5$5 faces, $8$8 edges and $5$5 vertices. One face is a square and the rest are triangles. Name it.

A square base with four triangular faces meeting at an apex is a square-based pyramid.

Answer: square-based pyramid.

Worked Example 6b