CalcOpenly

Volume and surface area calculator for 3D shapes

Volume, surface area, slant height and capacity in litres for 14 solids, including cylinders, cones, spheres, pyramids, prisms, tori and ellipsoids.

Updated Checked against 13 worked examples

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Volume
cm³
Volume: 282.7433 cm³
Shown to up to 4 decimal places, half-up
Total surface area
245.0442cm²
Lateral (curved) surface area
188.4956cm²
Base area
28.2743cm²
Capacity
0.282743L

The cylinder (r = 3 cm, h = 10 cm) holds 282.7433 cm³ — 0.2827 litres — and its outside surface is 245.0442 cm².

Cylinder (r = 3 cm, h = 10 cm)

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How it's calculated S

  1. Volume

    V=πr2h=π×32×10=282.743339 cm3V = \pi r^2 h = \pi \times 3^2 \times 10 = 282.743339\,\text{cm}^{3}

  2. Curved surface

    Slateral=2πrh=188.495559 cm2S_\text{lateral} = 2\pi r h = 188.495559\,\text{cm}^{2}

  3. Total surface area (with both ends)

    S=2πr(r+h)=245.044227 cm2S = 2\pi r(r + h) = 245.044227\,\text{cm}^{2}

  4. Capacity

    282.743339 cm3=0.28274334 L282.743339\,\text{cm}^{3} = 0.28274334\,\text{L}

    1 L = 1000 cm³ = 0.001 m³.

About the volume and surface area calculator

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.

Every result is exact except the surface of a general ellipsoid, which uses Knud Thomsen's approximation, within about 1.06%. The 3D model is drawn to the proportions entered; drag it to rotate.

Worked examples

Cylinder r = 3 cm, h = 10 cm

Solid
Cylinder
Radius
3 cm
Height
10 cm
Show results in
Centimetres (cm)
Volume
282.743339 cm³
Total surface area
245.044227 cm²
Lateral (curved) surface area
188.495559 cm²
Capacity
0.282743 L

Checked against: Python 3.8 math: 90*pi, 78*pi, 60*pi; 282.743… cm³ ÷ 1000 = L

Box 6 × 4 × 10 m

Solid
Cuboid (box)
Length
6 m
Width
4 m
Height
10 m
Show results in
Metres (m)
Volume
240 m³
Total surface area
248 m²
Space diagonal
12.328828 m
Capacity
240,000 L

Checked against: 6·4·10; 2(24 + 60 + 40); Python 3.8 math.sqrt(152); 1 m³ = 1000 L

Cone r = 3, h = 4 (slant 5)

Solid
Cone
Radius
3 cm
Height
4 cm
Show results in
Centimetres (cm)
Slant height
5 cm
Volume
37.699112 cm³
Lateral (curved) surface area
47.12389 cm²
Total surface area
75.398224 cm²

Checked against: 3-4-5 slant; Python 3.8 math: 12*pi, 15*pi, 24*pi

Frustum R = 5, r = 3, h = 4

Solid
Frustum of a cone
Bottom radius (R)
5 cm
Top radius (r)
3 cm
Height
4 cm
Show results in
Centimetres (cm)
Slant height
4.472136 cm
Volume
205.25072 cm³
Total surface area
219.211186 cm²

Checked against: Python 3.8 math: sqrt(20), pi*4*(25+15+9)/3, pi*8*sqrt(20) + 34*pi

Questions

How do you calculate the volume of a cylinder?

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.

How do you find the surface area of a cylinder?

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.

How many litres are in a cubic metre?

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.

What is the formula for the volume of a sphere?

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.

How do you calculate the volume of a cone or a pyramid?

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.

How accurate is the volume and surface area calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 13 worked examples whose answers come from independent sources; for example, “Cylinder r = 3 cm, h = 10 cm” is checked against Python 3.8 math: 90*pi, 78*pi, 60*pi; 282.743… cm³ ÷ 1000 = L.

Where does the method come from?

Zwillinger, D. (ed.) CRC Standard Mathematical Tables and Formulas, 33rd ed., §4.6 (solids); Weisstein, E. W. “Torus”, “Spherical Cap”, “Ellipsoid” — MathWorld; Thomsen, K. — ellipsoid surface approximation, as quoted in “Ellipsoid § Surface area”, Wikipedia.

About this calculator

Vsphere=43πr3,Vcone=13πr2hVfrustum=13πh(R2+Rr+r2),Vtorus=2π2Rr2\begin{gathered} V_\text{sphere} = \tfrac{4}{3}\pi r^3,\quad V_\text{cone} = \tfrac{1}{3}\pi r^2 h \\ V_\text{frustum} = \tfrac{1}{3}\pi h(R^2 + Rr + r^2),\quad V_\text{torus} = 2\pi^2 R r^2 \end{gathered}

Sources

  1. Zwillinger, D. (ed.) CRC Standard Mathematical Tables and Formulas, 33rd ed., §4.6 (solids)
  2. Weisstein, E. W. “Torus”, “Spherical Cap”, “Ellipsoid” — MathWorld
  3. Thomsen, K. — ellipsoid surface approximation, as quoted in “Ellipsoid § Surface area”, Wikipedia

Checked against references

13 worked examples with independently sourced answers ship with this calculator. They run in the test suite; you can run them here too.

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