The Surface Area and Volume Calculator calculates both the outside area and the inside capacity of a 3D shape in one step.
Surface Area and Volume Calculator
Surface area and volume together for a cube, box, cylinder, sphere, cone, hemisphere or square pyramid.
Surface area adds up the areas of every face (for example 2πr(r + h) for a cylinder), while volume multiplies the base area by the height or uses the shape’s own rule (for example πr²h). Choose a cube, rectangular prism, cylinder, sphere, cone, hemisphere or square pyramid and enter its dimensions.
Formula
| Shape | Surface area | Volume |
|---|---|---|
| Cube | A = 6a² | V = a³ |
| Rectangular prism | A = 2(lw + lh + wh) | V = lwh |
| Cylinder | A = 2πr(r + h) | V = πr²h |
| Sphere | A = 4πr² | V = (4/3)πr³ |
| Cone | A = πr(r + √(r² + h²)) | V = πr²h ÷ 3 |
| Hemisphere | A = 3πr² | V = (2/3)πr³ |
| Square pyramid | A = a² + 2a√(h² + (a/2)²) | V = a²h ÷ 3 |
Where:
- A = total surface area; V = volume
- a = side; l, w, h = length, width, height
- r = radius; h = vertical height
How the Calculation Works
Step 1: The calculator converts every dimension to the same unit.
Step 2: It applies the surface area formula, adding flat faces and curved surfaces together.
Step 3: It applies the volume formula. Prisms and cylinders use base area × height; cones and pyramids use one-third of that; spheres use (4/3)πr³.
Variables Explained
| Variable | Meaning | Unit |
|---|---|---|
| a, l, w, h, r | Dimensions of the shape | mm, cm, m, in, ft |
| A | Surface area | cm², m², in², ft² |
| V | Volume | cm³, m³, L, in³, ft³, gal |
Example Calculation
Cylinder
Given: radius = 3 cm, height = 10 cm
Surface area: A = 2 × 3.14159 × 3 × (3 + 10) ≈ 245.04 cm²
Volume: V = 3.14159 × 3² × 10 ≈ 282.74 cm³
Sphere
Given: radius = 5 cm
Surface area: A = 4 × 3.14159 × 25 ≈ 314.16 cm²
Volume: V = (4/3) × 3.14159 × 125 ≈ 523.60 cm³
Area and Volume Conversion Table
| Unit | Equivalent |
|---|---|
| 1 m² | 10,000 cm² |
| 1 ft² | 144 in² |
| 1 m³ | 1,000,000 cm³ = 1,000 L |
| 1 L | 1,000 cm³ |
| 1 ft³ | 1,728 in³ ≈ 7.48 US gal |
| 1 US gal | 231 in³ ≈ 3.785 L |
Shape Properties
How surface area and volume grow
If you double every dimension of a shape, its surface area becomes 4 times larger (2²) but its volume becomes 8 times larger (2³). That is why large objects have a smaller surface area compared with their volume.
Which shape holds the most?
For a given surface area, a sphere encloses the largest possible volume. That is why bubbles and water droplets are round.
Dimensions Explained
Use the radius for round shapes (half the diameter) and the vertical height for cones and pyramids. You can mix units: enter radius in inches and height in feet, and choose the result units you want.
Frequently Asked Questions
What is the difference between surface area and volume?
Surface area is the total area of the outside (square units). Volume is the space inside (cubic units).
How do you find the volume and surface area of a cube?
Volume = a³ and surface area = 6a². A cube with 5 cm sides has V = 125 cm³ and A = 150 cm².
How do you find the volume and surface area of a cylinder?
V = πr²h and A = 2πr² + 2πrh.
Can surface area and volume be equal?
Their numbers can match (a cube with 6-unit sides has A = 216 and V = 216), but they are different quantities in different units, so they are never truly “equal”.
How do I convert cubic centimeters to liters?
Divide by 1,000: 1,000 cm³ = 1 L. The calculator can also display volume directly in liters or gallons.