Solids
A three-dimensional figure (solid) has length, width and height. A prism has the same cross-section along its length. A pyramid has a polygon base and triangular faces meeting at an apex. A cylinder, a cone and a sphere have curved surfaces. Solids are drawn by sketching with dotted lines for hidden edges, and built from nets.
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Rectangular prism | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Square pyramid | 5 | 8 | 5 |
| Tetrahedron | 4 | 6 | 4 |
Surface area and volume
| Solid | Surface area | Volume |
|---|---|---|
| Cuboid | 2(lw + lh + wh) | lwh |
| Prism | 2 × base area + perimeter × length | base area × length |
| Cylinder | 2πr² + 2πrh | πr²h |
| Cone | πr² + πrl | ⅓πr²h |
| Pyramid | base + triangular faces | ⅓ × base area × h |
| Sphere | 4πr² | ⁴⁄₃πr³ |
Worked exampleA cylinder
A cylindrical tank has radius 7 cm and height 10 cm. Find its volume and curved surface area. (Use π = 22/7)
- Volume.
- Curved surface.
Answer: Volume 1 540 cm³, curved surface 440 cm².
Worked exampleA cone
A cone has radius 6 cm and height 8 cm. Find the slant height and the volume.
- Slant height l from the right triangle.
- Volume.
Answer: Slant height 10 cm, volume about 301.6 cm³.
Try itCylinder, cone and sphere
Choose the radius and height. Compare the volumes.
Key terms
- Prism
- A solid with a constant cross-section.
- Net
- A flat pattern that folds into a solid.
- Slant height
- The length along the side of a cone.
- Surface area
- The total area of all the faces.
Summary
- Learn the names and the numbers of faces, edges and vertices.
- Volume of prism = base area × length; of pyramid and cone = ⅓ base area × height.
- The volume of a sphere is ⁴⁄₃πr³ and its area 4πr².
- Nets show the surface of a solid laid flat.