3D Shapes: Volume & Surface Area
Volume and surface area of a cube, box, sphere, hemisphere, cylinder, cone, pyramid, triangular prism or frustum, with every step. The answer is exact with π where possible, in your unit and in litres, and the pictures show where the formulas come from: the base times the height, the cone as a third of a cylinder, and every surface unfolded into a net.
Volume and Surface Area
The volume is how much space a shape fills, in cubic units (cm³, m³): how much water it could hold. The surface area is the area of its outside, in square units (cm²): how much paint or wrapping paper it needs.
Three ideas give almost every formula. A shape with the same cross-section all the way up (box, cylinder, prism) has volume = base area × height. A shape that narrows to a point (cone, pyramid) holds exactly one third of that. And a sphere fills two thirds of the cylinder that just fits around it. For the surface area, imagine cutting the shape open and folding it flat into a net: the area of the net is the surface area.
Shape, Volume and Net
shapeStep-by-Step Solution
All the Formulas
| Shape | Volume | Surface area |
|---|---|---|
| Cube (edge a) | ||
| Box (l, w, h) | ||
| Sphere | ||
| Hemisphere | ||
| Cylinder | ||
| Cone | ||
| Square pyramid | ||
| Triangular prism | ||
| Frustum |
Sources & Verification
- H. S. M. Coxeter, Introduction to Geometry, 2nd ed., Wiley, 1969 — Ch. 10 (solids).
- D. Zwillinger (ed.), CRC Standard Mathematical Tables and Formulas, 33rd ed., CRC Press, 2018 — Sec. 4.6 (polyhedra) and 4.8 (cylinders, cones, spheres and frusta).
- Archimedes, On the Sphere and Cylinder, c. 225 BC — the sphere as two thirds of its circumscribed cylinder, in volume and in surface.
Frequently Asked Questions
How do you calculate the volume of a cylinder?
Multiply the area of the circular base by the height: V = πr²h. Example: r = 2 cm, h = 5 cm.
1. Base area: π × 2² = 4π ≈ 12.566 cm².
2. Times the height: 4π × 5 = 20π ≈ 62.83 cm³.
That is about 62.8 mL, because 1 cm³ = 1 mL.
How do you find the volume of a sphere?
V = ⁴⁄₃πr³. A football with a radius of 11 cm holds ⁴⁄₃ × π × 11³ ≈ 5575 cm³, about 5.6 litres. The ⁴⁄₃ comes from a discovery of Archimedes: a sphere fills exactly two thirds of the cylinder that just fits around it, and that cylinder holds πr² × 2r = 2πr³.
Why is the volume of a cone one third of a cylinder?
A cone with the same base and height as a cylinder holds exactly one third as much: V = ⅓πr²h. For r = 3 cm and h = 4 cm the cylinder holds 36π cm³ and the cone 12π ≈ 37.7 cm³. The same is true for a pyramid and the box around it: the Great Pyramid (base 230 m, height 146 m) is ⅓ × 230² × 146 ≈ 2.57 million m³.
How do you calculate the surface area of a cylinder?
Unroll it: the side becomes a rectangle 2πr wide (the circumference) and h high, and there are two circles for the top and bottom. So S = 2πr² + 2πrh. For r = 2 cm and h = 5 cm: 8π + 20π = 28π ≈ 87.96 cm².
How do you convert a volume to litres?
1 litre is 1000 cm³ (a cube of 10 × 10 × 10 cm), and 1 m³ is 1000 litres. A box of 40 × 30 × 20 cm holds 24,000 cm³ = 24 litres, about 6.34 US gallons. Measure in cm, divide the cm³ by 1000 and you have litres.
What is the difference between volume and surface area?
Volume is the space inside, in cubic units (cm³, m³): how much the shape holds. Surface area is the size of the outside, in square units (cm²): how much paint or wrapping it needs. Doubling every length makes the surface area 4 times and the volume 8 times as large.