Volume, surface area, slant height and capacity in litres for 14 solids, including cylinders, cones, spheres, pyramids, prisms, tori and ellipsoids.
By AdminDrafted with AI writing toolsUpdated Checked examples: 13
The cylinder (r = 3 cm, h = 10 cm) holds 282.7433 cm³ (0.2827 litres) and its outside surface is 245.0442 cm².
1 L = 1000 cm³ = 0.001 m³.
Choose one of 14 solids and enter its dimensions to get the volume and total surface area, plus the curved (lateral) surface, base area, slant height and space diagonal where the solid has them. The volume is also given as a capacity in litres, using 1 L = 1,000 cm³. The formulas are the standard ones, for example V = πr²h for a cylinder, V = ⅓πh(R² + Rr + r²) for a frustum and V = 2π²Rr² for a torus.
The default cylinder, 3 cm in radius and 10 cm tall, holds 282.743 cm³ (0.283 L) and has 245.044 cm² of surface, of which 188.496 cm² is the curved side. Volume sizes tanks, containers and concrete pours; surface area sizes paint, insulation and sheet material.
The formulas are exact except the surface of a general ellipsoid, which uses Knud Thomsen's approximation, within about 1.06%; numerical results are rounded for display. The 3D model is drawn to the proportions entered; drag it to rotate. In the overview formula, V₁, V₂, V₃ and V₄ denote sphere, cone, frustum and torus volumes respectively. Sₗ is lateral or curved area and B is base area.
Volume is the space inside a solid, in cubic units. For a cylinder or any prism it is the base area times the height, so a cylinder holds V = πr²h: a radius of 7 cm and a height of 10 cm give π × 7² × 10 = 490π ≈ 1,539.38 cm³, about 1.54 L.
In these formulas r is a radius, h the perpendicular height and S the total surface area.
A cone holds exactly one third of the cylinder with the same base and height, and a pyramid one third of the matching prism.
Curved or lateral surface area counts only the rounded side; a spherical cap and a torus each have their own formula. Total surface area adds the flat ends, which is what a closed can, tank or solid has.
For the cylinder with r = 7 cm and h = 10 cm, the curved side is 2πrh = 140π ≈ 439.82 cm². The two ends add 2πr² ≈ 307.88 cm², for a total of 2πr(r + h) = 238π ≈ 747.70 cm².
Answer keys that use π ≈ 22/7 give 1,540 cm³, 440 cm² and 748 cm² for the same cylinder. The method is identical; only the value of π differs.
A cone of radius 3 cm and height 4 cm has slant height √(9 + 16) = 5 cm. Its curved surface is πrl = 15π ≈ 47.12 cm², and the base brings the total to 24π ≈ 75.40 cm².
A hemisphere's curved surface is 2πr² and its total with the flat base is 3πr²: 56.55 cm² and 84.82 cm² for r = 3 cm. An open-topped tank has one end, so subtract one πr² from the cylinder's total.
All dimensions are in centimeters, volumes in cm³ and surfaces in cm². Radius means center to edge, half the diameter.
| Solid and dimensions | Volume, cm³ | Total surface area, cm² |
|---|---|---|
| Cube, edge 5 cm | 125 | 150 |
| Box, 6 × 4 × 10 cm | 240 | 248 |
| Sphere, radius 3 cm | 113.097336 | 113.097336 |
| Hemisphere, radius 3 cm | 56.548668 | 84.823002 |
| Cylinder, radius 3 and height 10 cm | 282.743339 | 245.044227 |
| Cone, radius 3 and height 4 cm | 37.699112 | 75.398224 |
| Frustum, radii 5 and 3, height 10 cm | 513.1268 | 363.118826 |
| Square pyramid, base edge 6 and height 4 cm | 48 | 96 |
| Equilateral triangular prism, edge 6 and height 10 cm | 155.884573 | 211.176915 |
| Regular hexagonal prism, edge 6 and height 10 cm | 935.307436 | 547.061487 |
| Torus, major radius 10 and tube radius 3 cm | 1776.528792 | 1184.352528 |
| Capsule, radius 3 and straight length 5 cm | 254.469005 | 207.345115 |
| Regular tetrahedron, edge 6 cm | 25.455844 | 62.353829 |
Doubling every dimension multiplies volume by 8 and surface by 4. The 5 cm cube holds 125 cm³ with 150 cm² of surface; a 10 cm cube holds 1,000 cm³ with 600 cm².
The sphere's two cells match because a sphere of radius 3 has both V and S equal to 36π, in cm³ and cm² respectively. The capsule is a 5 cm cylinder plus one whole sphere: 45π + 36π = 81π ≈ 254.47 cm³.
One liter is exactly 1,000 cm³, and one cubic meter is 1,000 liters. The 240 cm³ box in the table holds 0.24 L, and the 7 cm by 10 cm cylinder holds 1.539 L.
Cubic units convert with the cube of the length factor. A 2 × 3 × 4 ft box holds 24 ft³, which is 24 × 0.3048³ = 0.679604318208 m³, about 680 L. For a volume you already know, use the volume converter.
For a flat plan without depth, use the area calculator. For slabs, footings and bag counts, the concrete calculator adds waste allowances to the geometry.
Checked against: 90*pi, 78*pi, 60*pi; 282.743… cm³ ÷ 1000 = L
Checked against: 6·4·10; 2(24 + 60 + 40); 1 m³ = 1000 L
Checked against: Independent calculation
Checked against: sqrt(20), pi*4*(25+15+9)/3, pi*8*sqrt(20) + 34*pi
Multiply the area of the circular end by the height: V = πr²h. A radius of 3 cm and a height of 10 cm give 90π ≈ 282.74 cm³, or 0.283 L. Use the radius, not the diameter: a tank 1 m across and 1.5 m tall has r = 0.5 m and holds π × 0.25 × 1.5 ≈ 1.178 m³, which is 1,178 L.
Add the curved side, 2πrh, to the two circular ends, 2πr²: S = 2πr(r + h). For r = 3 cm and h = 10 cm the curved side is 60π ≈ 188.50 cm² and the total is 78π ≈ 245.04 cm². An open-topped tank has only one end, so subtract πr², here 28.27 cm², to get 216.77 cm².
1,000. A litre is exactly one cubic decimetre, 1,000 cm³, a definition fixed by the General Conference on Weights and Measures (CGPM) in 1964. Divide cubic centimetres by 1,000 to get litres. In US units, one gallon is exactly 231 in³, or 3.785411784 L, and one cubic foot is 28.316846592 L.
V = ⁴⁄₃πr³, which is two thirds of the cylinder that just encloses the sphere, as Archimedes showed. A sphere of radius 3 cm holds 36π ≈ 113.10 cm³, and a ball 1 m across holds 0.5236 m³, or 523.6 L. Volume grows with the cube of the radius, so doubling the diameter multiplies the volume by 8.
Take one third of the base area times the height: V = ⅓Bh. A cone of radius 3 and height 4 has V = ⅓π × 9 × 4 = 12π ≈ 37.70, and a square pyramid with a 6 × 6 base and height 4 has V = ⅓ × 36 × 4 = 48. A cone holds exactly one third of the cylinder with the same base and height.
This calculator includes 13 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.