Surface area formulas and volume formulas appear frequently in geometry, physics, engineering, and everyday problem solving. Whether you’re calculating pressure (force per area), density (mass per volume), or filling a container, knowing these formulas is essential.
📊 Tables for Quick Reference
Here are formulas for quick reference. Full descriptions follow.
Volume Formulas Table
| Shape | Volume Formula |
|---|---|
| Sphere | (4⁄3)πr³ |
| Cube | a³ |
| Rectangular Box | L×W×D |
| Prism | B×h |
| Cylinder | πr²×h |
| Pyramid | (1⁄3)B×h |
| Cone | (1⁄3)πr²×h |
Surface Area Formulas Table
| Shape | Surface Area Formula |
|---|---|
| Sphere | 4πr² |
| Cube | 6a² |
| Box | 2(LW + LD + WD) |
| Prism | 2B + Ph |
| Cylinder | 2πr² + 2πr×h |
| Pyramid | a² + 2a×s (square base) |
| Cone | πr² + πr×s |
🟠 Sphere Surface Area Formula and Sphere Volume Formula
A sphere is a perfectly round 3D shape where every point on the surface is equidistant from the center. The distance is the radius, r.
- Surface area = 4πr2
- Volume = (4⁄3)πr3
🧊 General Prism
A prism is a geometric shape consisting of a stack of identical base shapes stacked on top of each other to a depth d. This prism is a prism formed by a stack of triangles.
- Surface Area of a Prism = 2 × (Area of the base shape) + (Perimeter of base shape) × (d)
- Volume of a Prism = (Area of base shape) × d
Need to find the area and perimeter of the base shape? Use the Area Formulas and Perimeter Formulas.
🧱 Rectangular Prism (Box)
A box or rectangular prism is a stack of rectangles L long and W wide piled on top of each other to a depth of D.
Surface Area of a Box = Sum of the areas of each face of the box
- Surface Area of a Box = 2(L × W + L × D + W × D)
- Volume of a Box = L × W × D
🔲 Cube
A cube is a special case box where all the sides are the same length (a).
- Surface Area of a Cube = 6a2
- Volume of a Cube = a3
🥫 Cylinder
A cylinder is a prism where the base shape is a circle. It has bases of radius r and height h.
- Surface Area of a Cylinder = 2πr2 + 2πrh
- Volume of a Cylinder = πr2h
🔺 Square Pyramid
A pyramid is a solid shape consisting of a polygon base and triangular faces meeting at a common point above the base. A square pyramid is a pyramid where the base polygon is a square.
In the picture above, side a is the same length as side b. All of the face triangles are isosceles triangles meeting at a point h above the base.
For pyramids with identical face triangles (a = b = c)
The math is easier using the slant height (c) rather than the height (h):
- Surface Area = a² + 2as
- Volume = (1⁄3)a²h
🧁 Cone
A cone is a pyramid with a circular base with radius r and height h. Rind the side length or slant height (s) using the Pythagorean Theorem.
s2 = r2 + h2
or
s = √( r2 + h2 )
- Surface Area of a Cone = πr2 + πrs
- Volume of a Cone = 1⁄3( πr2h )
References
- Cook, James L. (1991). Conversion Factors. Oxford [England]: Oxford University Press. ISBN 0-19-856349-3.
- Deaver, James R. (1978). “Modeling Limits to Cell Size”. The American Biology Teacher. 40 (8): 502–504. doi:10.2307/4446369
- Lemons, Don S. (2017). A Student’s Guide to Dimensional Analysis. New York: Cambridge University Press. ISBN 978-1-107-16115-3.