Calculating area and volume comes up far more often than you might expect. How much wallpaper does this wall need? How large is that triangular plot of land? How many liters of water will fit in my aquarium? When the moment arrives, it is easy to draw a blank on the formula.
In this article, we cover how to find the area of squares, rectangles, triangles and circles, plus how to find the volume of cubes, cylinders and spheres, with formulas and worked examples. At the end, we show a free browser tool that does the math for you.
The difference between area and volume
Area is the amount of flat space a two-dimensional shape occupies. Its units use “squared”: square meters (m²) or square centimeters (cm²). Volume (also called capacity) is the amount of space a three-dimensional shape occupies, expressed in “cubed” units such as cubic meters (m³) or cubic centimeters (cm³).
Before you start, decide whether you need area or volume. Flat, two-dimensional figures use area; solids with thickness or height use volume. If you are unsure, ask “would water fill this?” - water fills solids, so it points to volume.
| Measure | Area | Volume |
|---|---|---|
| Meaning | Flat (2D) space | Space inside a solid (3D) |
| Units | m², cm², mm² | m³, cm³, mm³ |
| Everyday example | Floor, wall or land size | Aquarium water, concrete quantity |
Reading the units: read m² as "square meters" and cm³ as "cubic centimeters". Area and volume use different dimensions, so be careful not to mix them up.
How to find the area of 2D shapes
Here are the formulas for common plane shapes, with worked examples. Use the same unit for every dimension in a formula.
| Shape | Formula | Worked example |
|---|---|---|
| Square | side × side | 3cm × 3cm = 9cm² |
| Rectangle | width × height | 5cm × 3cm = 15cm² |
| Triangle | base × height ÷ 2 | 4cm × 3cm ÷ 2 = 6cm² |
| Parallelogram | base × height | 4cm × 3cm = 12cm² |
| Trapezoid | (top base + bottom base) × height ÷ 2 | (2cm + 6cm) × 3cm ÷ 2 = 12cm² |
| Circle | radius × radius × π | radius 5cm → 5 × 5 × 3.14 = 78.5cm² |
| Sector | radius × radius × π × angle ÷ 360 | radius 10cm, 90° → 78.5cm² |
Why the triangle formula divides by 2
The “÷ 2” in the triangle formula exists because two identical triangles, one flipped upside down, join to form a parallelogram. Think of it as the parallelogram area (base × height) cut in half, and the formula becomes easy to remember. The trapezoid works the same way: two identical trapezoids combine into a parallelogram, giving (top + bottom) × height ÷ 2.
In practice, the rectangle formula answers everyday questions like “How much wallpaper for a 3m by 4m wall?” - that is 3 × 4 = 12m². For a triangular plot with a 10m base and 6m height, the area is 10 × 6 ÷ 2 = 30m².
About π (pi): in formulas involving circles, π is commonly approximated as 3.14. Its exact value continues indefinitely at 3.14159..., so for serious precision it is safer to let a calculation tool handle it.
How to find the volume of 3D shapes
Next, the formulas for solid shapes. Volume measures the space inside a solid, so use it when you want to know how much water, oil or concrete will fill something.
| Shape | Formula | Worked example |
|---|---|---|
| Cube | side × side × side | 3cm → 3 × 3 × 3 = 27cm³ |
| Rectangular prism | width × length × height | 4cm × 3cm × 2cm = 24cm³ |
| Cylinder | radius × radius × π × height | radius 3cm, height 5cm → 3 × 3 × 3.14 × 5 = 141.3cm³ |
| Cone | radius × radius × π × height ÷ 3 | radius 3cm, height 6cm → 3 × 3 × 3.14 × 6 ÷ 3 = 56.52cm³ |
| Sphere | 4 ÷ 3 × π × radius × radius × radius | radius 3cm → 4 ÷ 3 × 3.14 × 27 = 113.04cm³ |
The connection between cylinders and cones
The “÷ 3” in the cone formula comes from the fact that a cone holds exactly one-third of the volume of a cylinder with the same base and height. So just take the cylinder formula (radius × radius × π × height) and divide by 3.
For a practical example, a concrete foundation that is 2m long, 2m wide and 2m high needs 2 × 2 × 2 = 8m³. Knowing the volume gives you a basis for estimating how much material you need.
Unit conversion and liters
If 1m = 100cm, what are 1m² and 1m³?
Since 1m = 100cm, one square meter equals 100cm × 100cm = 10,000cm², and one cubic meter equals 100cm × 100cm × 100cm = 1,000,000cm³. Working with mixed units in a single calculation will throw your answer off by orders of magnitude, so pick one unit and stick to it.
If you calculated in centimeters but need the answer in square meters, divide the cm² value by 10,000. For example, a floor that is 300cm by 400cm is 300 × 400 = 120,000cm² = 12m².
Measure water in liters
To find how much water an aquarium holds, convert the volume from cubic centimeters to liters: 1L = 1,000cm³.
Example: an aquarium that is 40cm long, 30cm wide and filled to a height of 20cm holds
40 × 30 × 20 = 24,000cm³ = 24L
Careful: mixing centimeters and meters in one formula shifts the answer by factors of 10 or 100. Before you start, convert every dimension to the same unit - 1m = 100cm and 1cm = 10mm.
Calculate area and volume for free in your browser
If mental math (or a paper formula sheet) is not working out, the Tools Hub Area & Volume Calculator handles it instantly in your browser. The numbers you enter are never sent to any server.
How to use it (3 steps)
Pick a shape
Open the Area & Volume Calculator and choose "Area (2D shapes)" or "Volume (3D shapes)", then select the shape you need.
Enter dimensions and unit
Fill in the fields shown for the shape (side, radius, height, and so on). The unit can be set to mm, cm, m, inches or feet.
Check the result
The area or volume appears instantly. In volume mode you also get the equivalent in liters, and the formula is displayed - handy for checking homework answers.
Tool featured in this article
Area & Volume Calculator
Supports 7 plane shapes and 5 solid shapes. Free, no sign-up, and everything runs in your browser - nothing is uploaded.
Summary
- Area measures flat space (m², cm²); volume measures inside a solid (m³, cm³)
- Triangle: base × height ÷ 2, circle: radius × radius × π, cone: one-third of its cylinder
- To find water in liters, divide the volume in cm³ by 1,000
- Match your units first - then use the free browser tool to check or speed up the math
The formulas stick better when you remember the why: a triangle is half a parallelogram, and a cone is one-third of its cylinder. For the actual calculation, the Tools Hub Area & Volume Calculator works right in your browser - perfect for homework checks and DIY estimates.
FAQ
What is the difference between area and volume units?
Area uses squared units such as m² and cm², while volume uses cubed units such as m³ and cm³. In conversion, 1m² = 10,000cm² and 1m³ = 1,000,000cm³.
How do I find the area of a circle?
Use “radius × radius × π (3.14)”. A circle with a radius of 5cm has an area of 5 × 5 × 3.14 = 78.5cm². If you need more precision, use 3.14159… for π or let a calculation tool do it.
How do I calculate how many liters of water an aquarium holds?
Multiply the fish tank’s length, width and water level (in cm), then divide by 1,000. For example, 40cm × 30cm × 20cm = 24,000cm³ = 24L.
What if I cannot remember the formulas?
Connect each formula to an image: a triangle is two copies of itself forming a parallelogram, and a cone is one-third of its cylinder. For the math itself, the Tools Hub Area & Volume Calculator does the work in your browser, which also makes a great answer checker.
