Volume and surface area for eight types of solids
Volume is the space inside a three-dimensional solid, measured in cubic units. Surface area is the area of its outside, measured in square units. This volume and surface area calculator supports a cube, rectangular prism, sphere, right circular cylinder, right circular cone, right triangular prism, right regular-polygon prism and right regular pyramid.
All models are closed ideal solids. Cylinder results include both ends; cone and pyramid results include their base. Lateral or curved area, where applicable, excludes the bases. A sphere has a whole curved surface rather than a separate base. For area of a flat regular polygon or supported quadrilateral, use the Polygon Area & Perimeter Calculator.
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Optional geometric solid models (paid link) can help identify bases, faces and height. No purchase is required.
How to calculate volume and total surface area
- Choose the solid and a shared length unit.
- Enter the required dimensions. Radius is half of diameter.
- Use perpendicular height for a cylinder, cone, regular prism or pyramid. Cone and pyramid slant heights are derived.
- For a right triangular prism, enter the two perpendicular legs of the triangular cross-section and the perpendicular prism length.
- For regular-polygon bases, enter the side count and one base side; the base must have equal sides and angles.
- Calculate and compare volume, total surface area, supporting measurements and steps.
Unit changes convert entered lengths rather than relabeling them. Hidden fields are disabled and ignored; editing clears the previous result. Reset selects cube and centimetres with blank length fields. Every dimension belongs to the same ideal solid; do not mix outside measurements with inside measurements.
Optional metric rulers (paid link) can help check a classroom model. Scientific notation is supported for lengths. Enter numbers without commas, fractions, arithmetic expressions or unit text in the numeric fields.
Volume, total surface area and lateral area formulas
Rectangular prism: V = lwh; total SA = 2(lw + lh + wh)
Sphere: V = 4πr³/3; total SA = 4πr²
Right cylinder: V = πr²h; total SA = 2πr² + 2πrh
Right cone: ℓ = √(r² + h²); V = πr²h/3; total SA = πr² + πrℓ
Right triangular prism: B = ab/2; c = √(a² + b²)
V = BL; total SA = 2B + (a + b + c)L
Regular base: P = ns; a = s/[2 tan(π/n)]; B = aP/2
Right regular prism: V = Bh; total SA = 2B + Ph
Right regular pyramid: ℓ = √(a² + h²); V = Bh/3; total SA = B + Pℓ/2
B denotes base area, P its perimeter, and ℓ the slant height. The symbol a denotes a right-triangle leg in that prism formula and the base apothem in the regular-base formulas. In a right prism the lateral faces are rectangles. In a right regular pyramid the apex is directly above the base centre and each triangular face has the same slant height.
For a cone the slant height uses radius; for a regular pyramid it uses the base apothem. Using the pyramid circumradius would find an edge to a vertex, not the triangular-face slant height required by Pℓ/2.
Worked cylinder, cone, prism and pyramid examples
A closed cylinder with radius 3 cm and height 5 cm has volume 45π ≈ 141.371669 cm³. Its two bases total 18π cm² and curved side is 30π cm², giving total surface area 48π ≈ 150.796447 cm².
A right cone with radius 3 cm and height 4 cm has slant height 5 cm. Volume is 12π ≈ 37.6991118 cm³. The curved side is 15π cm² and base is 9π cm², so total surface area is 24π ≈ 75.3982237 cm².
A right triangular prism with cross-section legs 3 cm and 4 cm has hypotenuse 5 cm and base area 6 cm². For length 10 cm, volume is 60 cm³ and total surface area is 2 × 6 + (3 + 4 + 5) × 10 = 132 cm².
For a right square pyramid, choose regular pyramid with n = 4, side 6 cm and vertical height 4 cm. Its base apothem is 3 cm and slant height is 5 cm. Volume is 48 cm³; lateral area is 60 cm²; total surface area is 96 cm². These are ideal geometry results, not construction dimensions.
Read cubic units and complete surface areas
The large result is volume. The total surface area card is a separate measurement and always includes every outer face of the closed model. A curved/lateral area card, if present, is not an alternative name for the total. For cone or pyramid modes, slant height describes the face; the entered vertical height describes the interior perpendicular distance to the base.
Doubling every length multiplies surface area by four and volume by eight. A change from centimetres to millimetres multiplies the displayed numbers by 100 for square units and by 1,000 for cubic units while preserving the same physical solid. For separate capacity conversions, use the Volume Converter.
Optional graph paper notebooks (paid link) can help record which unit power belongs to each quantity. Values are displayed to up to nine significant digits; floating-point calculations use unrounded intermediates and the usual numerical value of π. Input precision and real measurement uncertainty are not automatically inferred.
Closed-solid assumptions and supported input limits
Lengths must be positive and represent 10⁻⁹ to 10⁹ metres in the selected unit. Regular bases use an integer side count from 3 to 1,000. Derived dimensions may exceed input bounds. After Calculate, the diagram uses your dimensions, labels the calculated measurements and fits the projected solid uniformly into its box. Regular bases use your selected side count. Editing an input restores the reference sketch until you calculate again. Depth is projected in 3D views, and extremely thin features may merge on screen; the numeric labels remain authoritative.
The triangular-prism mode requires a right-triangle cross-section and a prism direction perpendicular to it. The regular pyramid requires a convex regular base and an apex above its centre. Oblique prisms/cones, nonregular or off-centre pyramids, frustums, hollow solids, open tanks, partial spheres and composite shapes are outside this page’s scope.
Optional geometry workbooks (paid link) can support practice with formulas and unit powers. No density, mass, fill allowance, coating thickness, packing loss or material cost is calculated. For project quantities use the existing Concrete Calculator or Swimming Pool Volume Calculator and read their assumptions. Do not use a closed surface-area result as an open-container design.
Volume & Surface Area Calculator FAQs
Cube, rectangular prism, sphere, right circular cylinder, right circular cone, right triangular prism, right regular-polygon prism and right regular pyramid.
Yes. Total surface area includes both circular ends plus the curved side. Curved area alone excludes the ends.
Enter perpendicular vertical height. The calculator derives cone slant height from radius and height, or regular-pyramid slant height from base apothem and height.
Yes. Select regular pyramid, enter n=4, one square base side and vertical height. The apex must be above the base centre.
No. This mode uses a right-triangle cross-section specified by its two perpendicular legs, with prism length perpendicular to that cross-section.
No. It models closed ideal solids. It does not subtract openings, wall thickness or internal cavities, combine shapes, or estimate material quantities.
Sources and methodology
Independently implemented standard geometry formulas, reviewed October 28, 2026. Sketches explain dimension conventions. Sources describe the mathematics; the linked organizations do not endorse this calculator.