# Area and volume formulas

The bottom layer of every take-off: plane areas and solid volumes with the symbols spelled out.

- Page: https://jianzhujisuan.com/en/geometry-formula-table
- Data checked: 2026-10-25

## Table

| Shape | Area / volume | Perimeter / surface |
|---|---|---|
| Rectangle | A = a × b | C = 2(a + b) |
| Triangle | A = b × h ÷ 2 | C = a + b + c |
| Trapezoid | A = (a + b) × h ÷ 2 | C = a + b + c + d |
| Parallelogram | A = b × h | C = 2(a + b) |
| Circle | A = πr² = πd²/4 | C = 2πr = πd |
| Sector | A = πr² × θ ÷ 360 | Arc L = πrθ ÷ 180 |
| Annulus | A = π(R² − r²) | — |
| Ellipse | A = πab | Perimeter ≈ π[3(a+b) − √((3a+b)(a+3b))] |
| Regular polygon (n sides) | A = nR²sin(2π/n) ÷ 2 | C = 2nRsin(π/n) |
| Cuboid | V = a × b × h | S = 2(ab + ah + bh) |
| Cylinder | V = πr²h | Side 2πrh, total 2πr(r + h) |
| Cone | V = πr²h ÷ 3 | Side πrl |
| Frustum of a cone | V = πh(R² + Rr + r²) ÷ 3 | Side π(R + r)l |
| Sphere | V = 4πr³ ÷ 3 | S = 4πr² |
| Pyramid | V = base area × h ÷ 3 | — |
| Frustum | V = h ÷ 6 × [S₁ + S₂ + 4S₀] | S₀ is the mid-section area |

## How to use

- Decompose first: break an irregular element into regular blocks, compute each and add. That beats forcing one formula.
- Cylindrical elements use V = πr²h with r the radius, not the diameter — the most common error on site.
- For pits and stockpiles use the frustum formula rather than a mean area, which understates by 5-10%.

## Common mistakes

- Substituting diameter for radius — the area comes out four times too large.
- Treating a sector as a triangle — fine below 60°, but more than 20% out above 90°.
- Using 3.14 for π — only 0.05%, but over hundreds of metres of pile that is cubic metres of concrete.

## Questions

### How do I calculate the area of a circle?

π times the radius squared. A 1 m diameter circle has a 0.5 m radius and an area of 0.785 m².

### How do I calculate the volume of a cylinder?

Base area times height, πr²h. A 0.5 m diameter, 10 m high cylinder is 1.96 m³.

### Why use the frustum formula for excavation?

Because side slopes make the top and bottom different sizes, and mean area × height understates consistently — by up to 10% depending on slope and depth.

## Related calculators

- [Circle and Sector Calculator](https://jianzhujisuan.com/en/circle-calc): Circle area and circumference plus sector, arc, chord and segment.
- [Cylinder Volume Calculator](https://jianzhujisuan.com/en/cylinder-volume): Volume, capacity, surface areas and weight for any cylinder.
- [Battered Pit Volume Calculator](https://jianzhujisuan.com/en/pit-volume): Top dimensions and frustum volume from base size, depth and batter.
- [Stair Calculator](https://jianzhujisuan.com/en/stair-calc): Riser count and height, run, slope length and slab area, with comfort check.