Area and volume formulas
The bottom layer of every take-off: plane areas and solid volumes with the symbols spelled out.
| 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 |
All values are computed from the stated formulas; ranges show the usual spread. Use measured data or the local code where it differs.
How to use this table
- 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%.
- Break the shape down first: irregular members split into regular blocks are far more reliable than one heroic formula.
- Height in a volume formula must be perpendicular to the base. On sloping members it is the vertical height, not the length along the slope — the wrong one systematically overstates.
- Surface area should count only the faces actually being finished. A formula gives the total surface, while waterproofing and painting rarely cover the same faces.
Common misuse
- 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.
- Computing triangle area as (a+b+c)/2. That is the semi-perimeter, not the area — use Heron's √(p(p−a)(p−b)(p−c)).
- Using 3.14 for π. At 1 m diameter the error is 0.05%, but at 10 m it is 0.04 m², and concrete volume amplifies it through the thickness.
- Computing sector area as arc length × radius. The correct form divides by two — the common error doubles it.
Calculate the actual number
Circle and Sector CalculatorCircle area and circumference plus sector, arc, chord and segment.Cylinder Volume CalculatorVolume, capacity, surface areas and weight for any cylinder.Battered Pit Volume CalculatorTop dimensions and frustum volume from base size, depth and batter.Stair CalculatorRiser count and height, run, slope length and slab area, with comfort check.
Other reference tables
FAQ
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.
How is Heron's formula used?
p = (a+b+c)/2, then area = √(p(p−a)(p−b)(p−c)). For sides 3, 4 and 5, p = 6 and the area is √(6×3×2×1) = 6, matching 3×4/2 for the right triangle.
How is the area of an ellipse calculated?
π × a × b, where a and b are the semi-axes, not the full axes. An ellipse with a 2 m major axis and a 1.2 m minor axis has an area of π × 1 × 0.6 = 1.885 m².
Why is a frustum not mean area × height?
Because the sides slope inward, the mid-section is smaller than the mean of the two end areas. The correct form is V = h/3 × (A₁ + A₂ + √(A₁A₂)); mean area × height overstates it, by about 8% when the end areas differ fourfold.
Results are estimating aids only and are not a design basis. Construction and final accounts must follow the drawings, the applicable standards and an approved method statement. Anything affecting structural safety, shoring or falsework must be checked by a suitably qualified engineer.