17.1 Properties of 2D and 3D shapes — Learn

Prerequisites: name familiar shapes and count their sides.

Objectives: use conventional geometry words, label a triangle, draw from a written description, and identify faces, surfaces, edges and vertices on common solids. This page covers G1 and G12.

Shape language and diagrams

A vertex is a corner. An edge is a straight boundary where two faces meet. A face is a flat surface; a curved solid has a curved surface. Parallel lines never meet. Perpendicular lines meet at a right angle. A polygon is a closed shape made from straight sides. A regular polygon has equal sides and equal angles. A shape has reflection symmetry if folding along a line matches both halves; it has rotational symmetry if a turn smaller than a full turn maps it onto itself.

Use capital letters for vertices. In triangle $ABC$, the side $AB$ joins vertices $A$ and $B$, and angle $A$ is written $\angle BAC$ or $\angle CAB$.

Worked example. A triangle has vertices $A$, $B$ and $C$. Its sides $AB$ and $AC$ are equal. Describe its symmetry.

• Equal sides meet at $A$.
• The line from $A$ to the midpoint of $BC$ splits the triangle into matching halves.
• So the triangle has one line of reflection symmetry.

Properties of solids

A cube has six square faces, twelve edges and eight vertices. A cuboid also has six rectangular faces, twelve edges and eight vertices. A triangular prism has two triangular faces and three rectangular faces. A cylinder has two flat circular surfaces and one curved surface, with no vertices. A cone has one flat circular surface, one curved surface and one vertex. A sphere has one curved surface and no edges or vertices. A pyramid has a polygon base and triangular faces meeting at an apex.

Worked example. A square-based pyramid has one square base and four triangular faces. It has five faces. The square base has four vertices and the apex adds one, so it has five vertices. Four base edges and four edges to the apex give eight edges.

Watch out. A curved surface is not a flat face. A cylinder has no vertices because its circular rims have no corners. Check by tracing each surface and each meeting point.

Progress check

The garden plan has a rectangular boundary. Name two pairs of parallel sides and state how many right angles it has. The centre lines divide it into four smaller rectangles.

You should now be able to

• use points, lines, vertices, edges, faces, planes and angle notation;
• describe regular polygons and reflection or rotation symmetry;
• draw a diagram from a written description;
• identify faces, surfaces, edges and vertices on common solids.