3D Solid Geometry, Explained
After reading this you can compute the volume, surface area and the handy extra lengths (slant height, diagonals, curved surfaces) for the seven classic solids, and know which formula fits the object in front of you.
What this calculator does
A solid is a region of space with a definite volume. This tool covers seven shapes you meet constantly: sphere, cube, rectangular box, cylinder, cone, square pyramid and torus. For each one it returns the two headline numbers, volume and surface area, plus a few derived lengths that the formulas produce along the way.
Here is the hook. A grain silo has a cylindrical body 3 m in radius and 7 m tall, topped by a cone of the same radius and 7 m of extra height. The cylinder holds \pi \cdot 3^2 \cdot 7 \approx 197.9 cubic metres. The cone adds \frac{1}{3}\pi \cdot 3^2 \cdot 7 \approx 65.97. Same base, same height, and the cone holds exactly one third of the cylinder. That factor of \frac{1}{3} is the single most useful fact in this article, and you will see why below.
When to use it, and when not
Use these formulas when your object really is one of the seven idealized shapes, or a simple sum of them. A soup can is a cylinder. A party hat is a cone. A concrete pillar with a domed top is a cylinder plus a hemisphere. Break a complex object into named pieces, compute each, then add or subtract.
Do not use them for shapes that only look regular. A slightly tapered can is a frustum, not a cylinder, and the straight cylinder formula will overstate its volume. A pyramid whose base is a rectangle rather than a square needs the general V = \frac{1}{3} B h with the correct base area B. If your problem is really two dimensional, for example the area of a floor plan, reach for the Polygon Area Calculator instead.
Every volume here scales with the cube of length and every surface area with the square. Double all the dimensions of a solid and its surface area grows by 2^2 = 4 while its volume grows by 2^3 = 8. That mismatch is why small animals lose heat fast and why crushed ice melts faster than a block.
The formulas and the intuition behind them
Start with the sphere, radius r.
The volume V and area A both hinge on r. A neat check: the derivative of the volume with respect to r is 4\pi r^2, which is exactly the surface area. Growing a sphere by a thin shell adds volume equal to area times thickness.
For a cylinder of radius r and height h:
The volume is base area times height. The surface area is two circular caps (2\pi r^2) plus the wall, which unrolls into a rectangle of width 2\pi r and height h.
For a cone with base radius r and height h, first the slant height:
The slant height \ell is the straight line from the rim to the apex, found with Pythagoras. The lateral surface is \pi r \ell, and the base disc adds \pi r^2. The factor \frac{1}{3} in the volume holds for every cone and pyramid.
The square pyramid, base edge a and height h, follows the same pattern:
Here \ell is the slant height of a triangular face, measured from the base edge midpoint to the apex. The four faces contribute 2a\ell in total.
The torus is the odd one out. With tube radius r and ring radius R (from the centre of the hole to the centre of the tube):
Both come from Pappus's theorem: spin a circle of area \pi r^2 and circumference 2\pi r around an axis, and each is multiplied by the distance its centroid travels, 2\pi R.
A worked example: the demo cone
Cone with r = 3, h = 7
This reproduces the calculator's demo data. Load it and press Calculate to check each line.
- Slant height: \ell = \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58} \approx 7.616.
- Volume: V = \frac{1}{3}\pi \cdot 3^2 \cdot 7 = \frac{1}{3}\pi \cdot 63 = 21\pi \approx 65.97.
- Base area: \pi \cdot 3^2 = 9\pi \approx 28.27.
- Lateral area: \pi \cdot 3 \cdot 7.616 \approx 71.79.
- Total surface area: 28.27 + 71.79 \approx 100.1.
So the cone holds about 65.97 cubic units and has a surface of about 100.1 square units. A cylinder with the same base and height would hold 63\pi \approx 197.9, three times as much, which is the one third factor again.
Reading and interpreting the results
Volume answers "how much fills it" and carries cubic units (cm³, m³, in³). Surface area answers "how much skin covers it" and carries square units. Keep them apart. If you paint a shape you care about area; if you fill it you care about volume.
The extra lengths matter in practice. The cone's slant height \ell \approx 7.616 is the length of the seam you would cut to make the cone from flat sheet, not the vertical height 7. For a box with edges a=4, b=5, c=6, the space diagonal is \sqrt{4^2 + 5^2 + 6^2} = \sqrt{77} \approx 8.775, which tells you the longest rod that fits inside.
| Shape | Inputs | Volume | Surface area |
|---|---|---|---|
| Sphere | r = 3 | 113.1 | 113.1 |
| Cube | a = 4 | 64.00 | 96.00 |
| Box | 4, 5, 6 | 120.0 | 148.0 |
| Cylinder | r = 3, h = 7 | 197.9 | 188.5 |
| Cone | r = 3, h = 7 | 65.97 | 100.1 |
| Square pyramid | a = 4, h = 7 | 37.33 | 74.51 |
Notice the sphere row: with r = 3 the volume 36\pi and area 36\pi are numerically equal. That happens only at r = 3, because \frac{4}{3}\pi r^3 = 4\pi r^2 forces r = 3. It is a coincidence of units, not a general truth.
Common mistakes
The most frequent error is using slant height where the formula wants vertical height, or the reverse. The cone volume uses the vertical height h = 7, giving 65.97. If you wrongly plug in \ell = 7.616 you get \frac{1}{3}\pi \cdot 9 \cdot 7.616 \approx 71.79, an 8.8 percent overstatement.
Two more traps. First, torus radii: R is measured to the centre of the tube, so the outer edge sits at R + r and the hole has radius R - r. A torus with R = 5 and r = 3 has a hole of radius 2 and an outer radius of 8. If you set r \gt R the tube self-intersects and the volume formula stops meaning what you think.
Second, mixing units. If one edge is in centimetres and another in millimetres, the product is nonsense. Convert everything to one unit before you start. Because volume scales as the cube, a units slip of 10 becomes a factor of 1000.
Related tools on this site
Solids rest on plane geometry, so several two dimensional tools pair well with this one. Use the Circle Calculator for the circular cross sections of cylinders, cones and spheres. The Right Triangle Calculator gives you slant heights and diagonals directly from Pythagoras. For a pyramid or prism on a many sided base, find that base area with the Regular Polygon Calculator, then multiply by height. When a slanted face needs an angle you do not have, the Trigonometry Calculator and the Triangle Solver fill the gap.
Frequently asked questions
Why is a cone one third of a cylinder?
Both share the base area \pi r^2 and height h, but the cone tapers to a point. Integrating the shrinking cross sections gives exactly \frac{1}{3}. At r = 3, h = 7 that is 65.97 versus 197.9.
What is the difference between lateral and total surface area?
Lateral surface area covers only the sides. Total surface area adds the flat ends. For the demo cone the lateral part is 71.79 and the base disc adds 28.27, giving 100.1 in total.
How do I find the volume of a shape not in this list?
Split it into named pieces. A capsule is a cylinder plus two hemispheres. A frustum is a large cone minus the small cone cut off the top. Compute each piece with these formulas and add or subtract.
Which radius goes where on a torus?
The ring radius R runs from the centre of the hole to the centre of the tube, and the tube radius r is the thickness of the tube itself. With R = 5, r = 3 the volume is 2\pi^2 \cdot 5 \cdot 9 \approx 888.3.
Do these formulas assume right, upright solids?
Yes. The cone and pyramid formulas assume the apex sits directly above the base centre. A slanted (oblique) cone has the same volume \frac{1}{3}\pi r^2 h if h is the perpendicular height, but its surface area differs and is not covered here.
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