Volume Formulas
Reference for volume formulas for cubes, prisms, cylinders, spheres, cones, and pyramids.
Includes the formula and a worked example for each shape.
Volume Formulas
Cube: V = s³
Rectangular Prism: V = l × w × h
Cylinder: V = π × r² × h
Sphere: V = (4/3) × π × r³
Cone: V = (1/3) × π × r² × h
Pyramid: V = (1/3) × base area × h
Rectangular Prism: V = l × w × h
Cylinder: V = π × r² × h
Sphere: V = (4/3) × π × r³
Cone: V = (1/3) × π × r² × h
Pyramid: V = (1/3) × base area × h
Variables
| Symbol | Meaning |
|---|---|
| V | Volume |
| s | Side length (cube) |
| l, w, h | Length, width, height |
| r | Radius |
| π | Pi, approximately 3.14159 |
Surface Area Formulas
Cube: SA = 6s²
Rectangular Prism: SA = 2(lw + lh + wh)
Cylinder: SA = 2πr² + 2πrh
Sphere: SA = 4πr²
Cone: SA = πr² + πr√(r² + h²)
Rectangular Prism: SA = 2(lw + lh + wh)
Cylinder: SA = 2πr² + 2πrh
Sphere: SA = 4πr²
Cone: SA = πr² + πr√(r² + h²)
Example 1 — Cylinder Volume
Find the volume of a cylinder with radius 5 cm and height 12 cm.
V = π × 5² × 12
V = π × 25 × 12 = 300π
V = 942.48 cm³
Example 2 — Sphere Volume
Find the volume of a sphere with radius 8 inches.
V = (4/3) × π × 8³
V = (4/3) × π × 512
V = 2,144.66 in³
Example 3 — Cone Volume
Find the volume of a cone with radius 6 m and height 10 m.
V = (1/3) × π × 6² × 10
V = (1/3) × π × 360
V = 376.99 m³
Quick Comparison
| Shape | Relationship |
|---|---|
| Cone vs Cylinder | Cone = ⅓ of a cylinder with same base and height |
| Pyramid vs Prism | Pyramid = ⅓ of a prism with same base and height |
| Sphere | Sphere = ⅔ of the cylinder that encloses it |
Key Notes
- Key formulas: Sphere: V = (4/3)πr³; cylinder: V = πr²h; cone: V = (1/3)πr²h; rectangular prism: V = lwh; pyramid: V = (1/3)Bh (B = base area). Note that cone = (1/3) × cylinder, and pyramid = (1/3) × prism — a cone fills its enclosing cylinder three times.
- Surface area companions: Sphere: A = 4πr²; cylinder (total): A = 2πr(r + h); cone (total): A = πr(r + l) where l = √(r² + h²) is slant height. Surface area and volume grow at different rates — surface area scales as r², volume as r³.
- Cavalieri's principle: If two solids have the same height and equal cross-sectional areas at every corresponding height, they have the same volume. This justifies the same formula for oblique and right versions of pyramids, cylinders, and cones.
- Scaling behavior: If all linear dimensions are scaled by factor k, volume scales by k³. Doubling all dimensions increases volume 8-fold. Surface area scales by k². This scaling difference explains why large animals need proportionally more structural support (bones) than small ones.
- Applications: Volume formulas are used in container design, material estimation, fluid capacity calculations, architectural planning, sports equipment standards (ball sizes), medicine (dosage based on organ volume), and manufacturing (material cost estimation).