0580

Geometry Toolkit

Geometry

Naming solids

  • A prism has the same cross-section all the way along, and it is named after that cross-section
    • A cube has a square cross-section and a cuboid a rectangular one
    • A triangular prism and a hexagonal prism are named the same way
A prism drawn end-on to show its cross-section, the same flat shape repeated all the way along the arrow marking its length
Source: Properties of 3D shapes by Save My Exams
  • A cylinder behaves like a prism whose cross-section is a circle
  • A pyramid stands on a base, with triangular sides rising to meet at a single point above it, and takes its name from that base
    • A pyramid on a triangular base is a tetrahedron
  • A cone is like a pyramid on a circular base, and a sphere is a ball
Seven solids each paired with the net that folds up to make it: a cube from 6 squares, a cuboid from 3 pairs of rectangles, a cylinder from 1 rectangle and 2 circles, a triangular prism from 3 rectangles and 2 triangles, a square-based pyramid from 1 square and 4 triangles, a tetrahedron from 4 triangles, and a cone from 1 circle and 1 sector
Source: Properties of 3D shapes by Save My Exams
  • A hemisphere is half a sphere, and a frustum is what remains when the top of a cone or pyramid is sliced off parallel to the base

Faces, edges and vertices

  • A face is a flat surface, an edge joins two faces, and a vertex is a corner; the plural of vertex is vertices
  • A curved surface, such as the side of a cylinder, counts as a surface rather than a face
SolidFacesEdgesVertices
Cube6 squares128
Cuboid6 rectangles, in 3 matching pairs128
Triangular prism2 triangles and 3 rectangles96
Square-based pyramid1 square and 4 triangles85
Tetrahedron4 triangles64
Cylinder2 circles, plus a curved surface20
Cone1 circle, plus a curved surface11
Spherecurved surface only00
  • For any solid with flat faces the three counts satisfy faces − edges + vertices = 2, which is a quick way to check a count
Six solids with their counts: cube 6 faces, 12 edges, 8 vertices; cuboid the same; cylinder 3 faces, 2 edges, 0 vertices; triangular prism 5 faces, 9 edges, 6 vertices; square-based pyramid 5 faces, 8 edges, 5 vertices; tetrahedron 4 faces, 6 edges, 4 vertices
Source: Properties of 3D shapes by Save My Exams

Nets

  • A net is the flat shape that folds up to make a solid, so it shows every face laid out
  • The area of a net equals the surface area of the solid, which is why nets are worth drawing for surface-area questions
A net of a cuboid coloured to show the matching pairs: two identical end faces, two identical top and bottom faces, and two identical side faces, folding to a box with three pairs of equal rectangles
Source: Properties of 3D shapes by Save My Exams
  • A solid has more than one possible net, and a cube has eleven, so an arrangement different from yours may still be correct
One net of a cube: a row of four squares with one more square joined above the second and one more below it, forming a cross
Source: Properties of 3D shapes by Save My Exams
  • The net of a cylinder is two circles and a rectangle whose length equals the circumference of those circles
A cylinder of radius r and height h beside its net: a rectangle of height h whose length is the circumference 2 pi r, with a circle of radius r joined at the top and another at the bottom
Source: Properties of 3D shapes by Save My Exams
  • Check a proposed net by asking whether the faces would overlap or leave a gap when folded
A square-based pyramid with a base edge of 15 cm and a slant height of 23 cm, beside its net: the 15 cm square in the middle with a triangle of height 23 cm folded out from each of its four sides
Source: Properties of 3D shapes by Save My Exams
Worked example

Counting faces, edges and vertices

Write down the number of faces, edges and vertices of a square-based pyramid and of a triangular prism.

Solution:

  • For the square-based pyramid, count the flat surfaces first
  • One square base and four triangles gives 5 faces
  • The base has 4 edges, and 4 more rise from its corners to the apex, giving 8 edges
  • The 4 corners of the base plus the apex give 5 vertices
  • For the triangular prism, the two triangular ends plus three rectangles give 5 faces
  • Three edges round each triangle plus three joining them gives 9 edges
  • Three corners on each triangle gives 6 vertices
  • Check each with Euler's relationship, faces + vertices − edges = 2: 5 + 5 − 8 = 2, and 5 + 6 − 9 = 2