ShapesSurface area and volume of solids
A solid has two things worth measuring. Its surface area is the total area of its outside, the amount of paper needed to wrap it. Its volume is the space it takes up, or for a container, how much it holds. This page explains the terms used in the chapter and lists the formulas side by side.
The words
- Lateral surface area (LSA) is the area of the side faces only, leaving out the top and the bottom. For a room it is the four walls.
- Curved surface area (CSA) is the same idea for round solids: the curved part only, without the flat circular ends.
- Total surface area (TSA) is every face added together.
- Volume is measured in cubic units: cm³, m³.
- Capacity is the volume a hollow container can hold, usually given in litres. One litre is 1000 cm³.
All the formulas
| Shape | Formulas |
|---|---|
| Cube | Volume = a3 Total surface area = 6a2 Lateral surface area = 4a2 Diagonal = a√3 |
| Cuboid | Volume = l × b × h Total surface area = 2(lb + bh + hl) Lateral surface area = 2h(l + b) Diagonal = √(l2 + b2 + h2) |
| Cylinder | Volume = πr2h Curved surface area = 2πrh Total surface area = 2πr(r + h) |
| Cone | Slant height l = √(r2 + h2) Volume = ⅓ πr2h Curved surface area = πrl Total surface area = πr(r + l) |
| Sphere | Volume = (4 ⁄ 3) πr3 Surface area = 4πr2 |
| Hemisphere | Volume = (2 ⁄ 3) πr3 Curved surface area = 2πr2 Total surface area = 3πr2 |
| Square pyramid | Slant height l = √(h2 + a2 ⁄ 4) Volume = ⅓ a2h Lateral surface area = 2al Total surface area = a2 + 2al |
| Triangular prism | End area by Heron's formula Volume = end area × L Lateral surface area = (a + b + c) × L Total surface area = lateral area + 2 × end area |
| Frustum of a cone | Slant height l = √(h2 + (R − r)2) Volume = ⅓ πh(R2 + Rr + r2) Curved surface area = π(R + r)l Total surface area = π(R + r)l + π(R2 + r2) |
| Hollow cylinder (pipe) | Volume = π(R2 − r2)h Curved surface area = 2π(R + r)h Total surface area = 2π(R + r)h + 2π(R2 − r2) Capacity = πr2h |
In the table, r is a radius, h a height and l a slant height.
Two ideas that explain most of them
Prisms and cylinders: base area × height. If a solid has the same cross-section all the way up, its volume is the area of that cross-section times the height. That covers the cube, the cuboid, every prism and the cylinder.
Pyramids and cones: one third of that. A solid that narrows evenly to a point has one third of the volume of the prism or cylinder with the same base and height.
Which surface does the question want?
- Painting the four walls of a room: lateral surface area, minus doors and windows.
- Painting walls and ceiling: lateral surface plus one base.
- Tin sheet for a closed tin: total surface area.
- Tin sheet for an open tank: curved surface plus one base.
- Canvas for a conical tent: curved surface only, since a tent has no floor.
- Label round a bottle: curved surface.
Reading the question for "open", "closed", "hollow" and "solid" settles most surface area sums.
Melting and recasting
When a solid is melted and made into another shape, the surface area changes but the volume stays the same. Set the two volumes equal and solve for the unknown. This one idea answers a whole family of exam questions.
Scaling
Double every length of a solid: its surface area becomes 4 times and its volume 8 times as large. That is why a large pot holds much more than twice a pot of half its height.
See each solid's page from the home page, try the calculator, or read the solved word problems.