Area / Volume Calculator
Computes area for rectangles, circles, and triangles, and volume for boxes, cylinders, and spheres. The formulas are elementary; the part that consistently causes expensive mistakes is how these quantities scale, because area and volume do not grow in proportion to length.
How to use it
- Choose the shape.
- Enter the dimensions the shape needs.
- Read the area or volume with its unit.
The formulas
Area: a rectangle is length times width; a circle is pi times radius squared; a triangle is half base times height, where the height is the perpendicular distance to the base and not the length of a sloping side.
Volume: a box is length times width times height; a cylinder is pi times radius squared times height; a sphere is four thirds pi times radius cubed.
The single most common input error is using diameter where the formula wants radius. Because the radius is squared or cubed, that mistake produces a result four times too large for a circle area and eight times too large for a sphere volume. Any figure that seems implausibly large is worth checking here first.
The square-cube law
Doubling every linear dimension of an object multiplies its area by four and its volume by eight. This is the square-cube law, and its consequences are far-reaching.
A worked case: a 1 metre cube has 6 square metres of surface and 1 cubic metre of volume, a ratio of 6 to 1. A 2 metre cube has 24 square metres and 8 cubic metres, a ratio of 3 to 1. The surface area per unit volume halves.
That ratio governs a great deal of physical behaviour. Large objects lose heat more slowly than small ones of the same shape, which is why small animals eat proportionally more and why a large concrete pour generates enough internal heat to require thermal management. It is why a scale model of a structure cannot be tested by simply scaling the loads, since weight grows with volume while strength grows with cross-sectional area. And it is why doubling a tank capacity requires only about 1.6 times the material, which is the economy of scale behind large storage vessels.
The practical consequence for estimating is that a scale factor cannot be applied to a quantity of a different dimension. Doubling the dimensions of a room does not double the paint required — it quadruples it. Doubling a pool does not double the water — it multiplies it by eight.
Turning volume into material
A volume figure is the start of a material estimate, not the end of one.
- Concrete is ordered in cubic yards in the United States and cubic metres elsewhere, and a slab volume is length times width times thickness with the thickness in the same unit. A 10 by 12 foot slab at 4 inches is 40 cubic feet, or about 1.48 cubic yards. Order 5 to 10 percent extra for subgrade irregularity and spillage.
- Paint coverage is typically 350 to 400 square feet per gallon on a smooth sealed surface, and considerably less on textured, porous, or unprimed substrates. Two coats doubles the requirement.
- Mulch and gravel are sold by volume but spread by depth, so the calculation is area times depth. Both settle, so allow extra.
- Water weight is a frequent oversight in structural terms: one cubic metre is 1,000 kg, and one cubic foot is about 62.4 pounds. A modest aquarium or hot tub is a substantial point load.
- Excavated soil expands, commonly 20 to 30 percent, so the volume to be hauled away exceeds the hole volume.
Irregular shapes
Real spaces are rarely a single primitive. The reliable method is decomposition: divide the shape into rectangles, triangles, and partial circles, compute each, and add or subtract.
An L-shaped room is two rectangles. A room with a bay window is a rectangle plus a half circle or trapezoid. A wall with openings is the gross rectangle minus each door and window, which for paint or plaster estimating is worth doing rather than ignoring — three windows and two doors can easily be 15 percent of a wall area.
For genuinely curved or irregular boundaries, the trapezoidal method works well enough: take measurements at regular intervals across the shape, treat each strip as a trapezoid, and sum. Accuracy improves with more intervals, and for most practical purposes a handful is sufficient.
At a glance
| Areas | Rectangle, circle, triangle |
|---|---|
| Volumes | Box, cylinder, sphere |
| Circles | Radius, not diameter |
| Scaling | Area scales with the square, volume with the cube |
Frequently asked questions
Why is my circle answer four times too big?
You almost certainly entered diameter where radius was wanted. The radius is squared, so the error is a factor of four — and eight for a sphere volume.
If I double the room size, do I need twice the paint?
No, four times. Area scales with the square of linear dimensions. Volume scales with the cube, so doubling a pool multiplies the water by eight.
How do I calculate an L-shaped room?
Split it into two rectangles, compute each, and add. Subtract openings for wall area estimates.
How much concrete should I order?
The calculated volume plus 5 to 10 percent for subgrade irregularity and spillage. Convert cubic feet to cubic yards by dividing by 27.
Read more
Engineering and vehicle math — Squared and cubed relationships explain most of it: heat, scaling, and why your truck runs out of payload before tow rating.