GCSE Maths · Volume, Surface Area and 3D Shapes · Higher · Summary
GCSE Maths Volume, Surface Area and 3D Shapes (Higher): summary and key words
Part of the GCSE Maths volume, surface area and 3d shapes chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page
Chapter summary
| Skill | Method |
|---|---|
| Properties of solids | Face = flat surface, edge = where two faces meet, vertex = corner. Euler: V − E + F = 2. n-sided prism: n + 2 faces, 3n edges, 2n vertices. n-sided pyramid: n + 1 faces, 2n edges, n + 1 vertices |
| Nets | A net folds to make the solid. Cube: 6 squares (11 different nets). Cylinder: rectangle 2πr by h and two circles. Cone (H): sector of radius l and angle 360r ÷ l |
| Plans and elevations | Plan = from above, front elevation = from the front, side elevation = from the side (front on the left). Use the tallest stack in each column or row |
| Volume of a cuboid | length × width × height. Cube: side3 |
| Units of volume | 1 ml = 1 cm3. 1 litre = 1000 cm3. 1 m3 = 1000 litres = 1,000,000 cm3 |
| Volume of a prism | area of cross-section × length (use the perpendicular height) |
| Cylinder | V = πr2h. Total surface area = 2πr2 + 2πrh. Curved surface = 2πrh |
| Density | density = mass ÷ volume; mass = density × volume; volume = mass ÷ density |
| Surface area of a cuboid or prism | Cuboid: 2lw + 2lh + 2wh. Prism: 2 × (cross-section area) + (perimeter of the cross-section) × length |
| Pyramid and cone (H) | Pyramid V = ⅓ × base area × vertical height. Cone V = ⅓πr2h, curved surface πrl, total πrl + πr2. l2 = r2 + h2 |
| Sphere and hemisphere (H) | Sphere V = &frac43;πr3, surface 4πr2. Hemisphere V = ⅔πr3, curved 2πr2, total 3πr2 |
| Composite solids and frustums (H) | Split into standard solids; add volumes, subtract holes; count only outside surfaces. Frustum = large cone − small cone (similar triangles) |
| Similar solids (H) | Length scale factor k: area × k2, volume (and mass) × k3 |
Can you…?
- I can name solids and count their faces, edges and vertices, and use Euler’s formula.
- I can recognise nets of cubes, cuboids, prisms, pyramids and cylinders.
- I can draw the plan and elevations of a solid made of cubes and of standard solids.
- I can work out the volume of a cuboid, prism and cylinder, and convert between cm3, litres and m3.
- I can use density = mass ÷ volume.
- I can work out the surface area of a cuboid, prism and cylinder.
- Higher: I can work out the volume and surface area of pyramids, cones, spheres and hemispheres.
- Higher: I can solve composite-solid, frustum, working-backwards and similar-solid problems.
Key terms
- Solid (3D shape)
- A shape with length, width and height.
- Face
- A flat surface of a solid.
- Edge
- A line where two faces of a solid meet.
- Vertex
- A corner of a solid (plural: vertices).
- Prism
- A solid with the same cross-section all along its length.
- Cross-section
- The face you see when you slice straight through a prism.
- Pyramid
- A solid with a polygon base and triangular faces meeting at an apex.
- Net
- A flat pattern that folds up to make a solid.
- Plan
- The view of a solid from above.
- Elevation
- A view of a solid from the front or the side.
- Volume
- The amount of space inside a solid, in cubic units.
- Capacity
- The amount a container can hold, in litres or millilitres.
- Surface area
- The total area of all the faces of a solid.
- Density
- The mass of a material in each unit of volume.
- Slant height
- The length along a sloping surface of a cone or pyramid.
- Frustum
- A cone with the top cut off by a plane parallel to the base.
- Similar solids
- Solids that are enlargements of each other, with all lengths in the same ratio.