In-depth
Area and Volume Calculator: All Geometric Formulas
Calculating the area of a surface or the volume of a solid is a recurring need: from calculating the square meters of a room to the amount of soil to fill a planter, from the area of a plot of land to the volume of a tank. This calculator brings together the formulas of all major geometric shapes, both flat and solid.
Formulas for flat shapes (area)
| Shape | Formula | Variables |
|---|---|---|
| Square | A = l² | l = side |
| Rectangle | A = b × h | b = base, h = height |
| Triangle | A = (b × h) ÷ 2 | b = base, h = height |
| Circle | A = π × r² | r = radius |
| Trapezoid | A = (B + b) × h ÷ 2 | B = longer base, b = shorter base |
| Parallelogram | A = b × h | b = base, h = height |
| Rhombus | A = (d₁ × d₂) ÷ 2 | d₁, d₂ = diagonals |
| Ellipse | A = π × a × b | a, b = semi-axes |
| Regular hexagon | A = (3√3 ÷ 2) × l² | l = side |
Heron's formula for triangles
When you don't know the height of a triangle but you have the three sides (a, b, c), you can use Heron's formula:
- Calculate the semi-perimeter: s = (a + b + c) ÷ 2
- Calculate the area: A = √[s × (s − a) × (s − b) × (s − c)]
Example: a triangle with sides 5, 7, and 8 cm. Semi-perimeter: s = (5 + 7 + 8) ÷ 2 = 10. Area = √(10 × 5 × 3 × 2) = √300 ≈ 17.32 cm².
Formulas for solids (volume)
| Solid | Volume | Total surface area |
|---|---|---|
| Cube | V = l³ | S = 6l² |
| Rectangular prism | V = l × w × h | S = 2(lw + lh + wh) |
| Cylinder | V = π × r² × h | S = 2πr(r + h) |
| Sphere | V = (4/3)πr³ | S = 4πr² |
| Cone | V = (1/3)πr²h | S = πr(r + √(r² + h²)) |
| Pyramid | V = (1/3) × A_base × h | Depends on the base |
| Truncated cone | V = (πh/3)(R² + Rr + r²) | R = larger radius, r = smaller |
Practical calculation examples
Area of a room
A rectangular room measures 4.5 m × 3.8 m. The area is: 4.5 × 3.8 = 17.1 m². To calculate the tiles needed (e.g., 30×30 cm = 0.09 m² each): 17.1 ÷ 0.09 = 190 tiles (plus 10% waste = about 209).
Volume of a cylindrical tank
A tank has a radius of 0.5 m and a height of 1.2 m. Volume: π × 0.5² × 1.2 = π × 0.3 ≈ 0.942 m³ = 942 liters.
Surface area of a sphere (ball)
A soccer ball has a diameter of 22 cm, so a radius of 11 cm. Surface area: 4π × 11² = 4π × 121 ≈ 1,520.5 cm².
Units of measurement for area and volume
| Quantity | Unit | Equivalence |
|---|---|---|
| Area | 1 m² | 10,000 cm² |
| Area | 1 hectare (ha) | 10,000 m² |
| Area | 1 km² | 1,000,000 m² |
| Volume | 1 m³ | 1,000 liters |
| Volume | 1 liter | 1,000 cm³ (ml) |
| Volume | 1 cm³ | 1 ml |
Perimeter and circumference
The calculator also provides the perimeter of flat shapes:
- Square: P = 4l
- Rectangle: P = 2(b + h)
- Circle (circumference): C = 2πr = πd
- Triangle: P = a + b + c
How to use the calculator
Select the desired geometric shape (flat or solid), enter the required dimensions in the corresponding fields, and press "Calculate." You will immediately get the area (or volume), the perimeter (or total surface area), and, where applicable, other useful measurements such as the diagonal or apothem. You can change the units of measurement to suit your needs.
