Calculating the surface area and volume of geometric solids is a skill needed in construction, architecture, physics and everyday life. Whether you want to paint a room, plan a pool, or calculate a tank's capacity, you always need formulas for solids. Below is a complete set of formulas with worked examples.
Cube
A cube is a rectangular box with all sides equal (edge length = a):
- Surface area:
A = 6a² - Volume:
V = a³
Example: a cube with a 5 cm edge has surface area A = 6 · 25 = 150 cm² and volume V = 125 cm³.
Rectangular box (cuboid)
A cuboid with dimensions a × b × c:
- Surface area:
A = 2(ab + bc + ac) - Volume:
V = a · b · c
Example: a box measuring 30 × 20 × 10 cm: V = 6000 cm³ = 6 liters.
Cylinder
A cylinder with base radius r and height h:
- Total surface area:
A = 2πr² + 2πrh = 2πr(r + h) - Lateral surface area:
Alateral = 2πrh - Volume:
V = πr²h
Example: a cylinder with r=10 cm, h=30 cm: V = π · 100 · 30 ≈ 9425 cm³ ≈ 9.4 liters.
Cone
A cone with base radius r, height h and slant height l (where l² = r² + h²):
- Total surface area:
A = πr² + πrl = πr(r + l) - Volume:
V = (1/3)πr²h
A cone's volume is exactly 1/3 the volume of a cylinder with the same dimensions — an elegant geometric property.
Sphere
A sphere with radius r:
- Surface area:
A = 4πr² - Volume:
V = (4/3)πr³
Example: a ball with a 22 cm diameter (r=11 cm): V = (4/3)π · 1331 ≈ 5575 cm³.
Regular prism
A prism with a regular n-sided polygon base, base side length a and height h:
- Base area:
Abase = (n · a² / 4) · cot(π/n) - Volume:
V = Abase · h
For a triangular prism (n=3) or a hexagonal prism (n=6), these formulas simplify to their well-known forms.
Pyramid
A pyramid with any polygonal base of area Abase and height h:
- Volume:
V = (1/3) · Abase · h
The one-third rule holds for every pyramid, regardless of the base's shape.
Sphere vs. cylinder vs. cone — comparing volumes
For the same radius r and a height h = 2r (so the sphere fits exactly inside the cylinder):
- Cylinder: V = πr² · 2r = 2πr³
- Sphere: V = (4/3)πr³ ≈ 1.333πr³
- Cone: V = (1/3)πr² · 2r = (2/3)πr³ ≈ 0.667πr³
Archimedes was famously proud of discovering that the ratio of a sphere's volume to a cylinder's is 2:3.
Practical applications
- Painting walls — the surface area of a room's walls and ceiling (a cuboid minus the floor).
- Tank capacity — cylindrical tanks for fuel, water, gas.
- Garden soil — the volume of a rectangular or prism-shaped area when planning earthworks.
- Concrete — the volume of foundations and concrete slabs.
Use our geometric solids calculator for quick calculations without memorizing formulas.
FAQ — questions about geometric solids
What's the formula for a sphere's volume?
A sphere's volume = (4/3) · π · r³, where r is the radius. For diameter d: r = d/2. Example: a sphere with a 10 cm diameter has V = (4/3)π · 125 ≈ 523.6 cm³.
How do you calculate a cylinder's surface area?
A cylinder's total surface area = 2πr(r + h), where r is the base radius and h is the height. It includes two bases (circles) and the lateral surface (a rectangle rolled into a tube).
What's the difference between surface area and volume?
Surface area measures a solid's outer shell (units: cm², m²). Volume measures the space inside it (units: cm³, m³, liters). E.g. to paint a wall you need area; to fill a pool, you need volume.
How many liters fit in a cylinder with a 1 m diameter and 1 m height?
V = π · (0.5)² · 1 = π · 0.25 ≈ 0.785 m³ = 785 liters. Remember: 1 m³ = 1000 liters.
How do you calculate a cone's slant height?
Slant height l = √(r² + h²), where r is the base radius and h is the height. It's simply the diagonal of a right triangle with legs r and h.
What's the formula for a hexagonal prism's base area?
The area of a regular hexagon with side a: Abase = (3√3/2) · a². Total surface area of the prism: A = 2 · Abase + 6 · a · h, where h is the height.
How much soil is needed for a cone-shaped mound?
A cone's volume V = (1/3)πr²h. E.g. a mound with base radius 3 m and height 2 m: V = (1/3)π · 9 · 2 ≈ 18.85 m³. Soil density is about 1.2-1.5 t/m³, so the mound weighs roughly 22-28 tons.
How do you calculate the volume of an irregular solid?
Using Archimedes' method: submerge the solid in water and measure the volume of water displaced. Engineering calculations use numerical integration or CAD software.
What's the SI unit of volume?
The base SI unit of volume is the cubic meter (m³). A liter (L) is 0.001 m³ = 1 dm³. 1 cm³ = 1 mL. Conversion: 1 m³ = 1000 L = 1,000,000 cm³.
How do you check a volume calculation?
Check the units — volume must have cubic units (cm³, m³). Verify by comparison: does the result seem reasonable? E.g. a pool 10×5×2 m is 100 m³ = 100,000 liters — exactly the amount of water needed to fill it.