One idea covers most shapes
Most volume formulas are the same formula in disguise: base area times height. Give a box a rectangular base and you get length x width x height. Swap in a circle and the cylinder is pi x r^2 x height. With a triangle on the end, a prism is half of base x height, times the length.
That works whenever the cross-section stays the same all the way up. Slice a can of soup anywhere and you get the same circle. Once you see that, you don't need to memorize the prism and cylinder formulas separately.
Formula notes
Prism or cylinder: V = base area x heightBox: V = l x w x hCylinder: V = pi x r^2 x h
Shapes that come to a point take one third
Cones and pyramids shrink to a point, so they hold less. The exact amount is one third of the matching cylinder or box. A pyramid with a 3 x 3 m base and 4 m height is 3 x 3 x 4 / 3 = 12 cubic meters. The box around it would be 36.
Calculus proves it by adding up thin slices, but you can test it in a kitchen. Fill a cone-shaped cup with water three times and it fills a cylinder of the same height and width.
Formula notes
Cone: V = pi x r^2 x h / 3Pyramid: V = l x w x h / 3
Spheres and the two-thirds link
A sphere is 4/3 x pi x r^3. That number looks random until you compare it with a cylinder that just fits around the ball. That cylinder has radius r and height 2r, so its volume is 2 x pi x r^3. The sphere is exactly two thirds of it.
A hemisphere is half a sphere, or 2/3 x pi x r^3. Because radius is cubed, doubling the size of a ball multiplies its volume by eight.
Formula notes
Sphere: V = 4 x pi x r^3 / 3Hemisphere: V = 2 x pi x r^3 / 3
Shapes in between: frustums and tubes
A bucket or a lampshade is a cone with its tip cut off, called a frustum. Its formula keeps both radii: pi x h x (R^2 + R x r + r^2) / 3. A bucket 10 inches tall, 6 inches in radius at the base and 4 at the top, holds about 795.87 cubic inches, or 3.45 U.S. gallons.
A pipe or tube wall is a big cylinder minus a small one: pi x (R^2 - r^2) x h. That gives you the material, which is what you'd need for weight. The space inside is just the inner cylinder.
Mistakes that wreck the answer
Using the diameter as the radius is the big one. It makes a cylinder or cone four times too large and a sphere eight times too large. Using the slant height of a cone or pyramid is the next. The formula wants the height straight up from the base to the tip.
Mixed units cause the rest. If length is in feet and depth is in inches, convert first. And remember that outside measurements include the walls, so a container's real capacity is smaller.
Our volume calculator handles the unit conversion and shows the formula it used, which makes these slips easy to spot.
Turning cubic units into liquid
One liter is a cubic decimeter, so 1 cubic meter is 1,000 liters. In U.S. units, a cubic foot holds about 7.48 U.S. gallons and a gallon is exactly 231 cubic inches. An Imperial gallon is bigger, at 4.54609 liters, so check which gallon your source means before you compare numbers.
Frequently asked questions
What is the formula for volume?
For any shape with a constant cross-section, volume is base area times height. Shapes that taper to a point, like cones and pyramids, use one third of that. Spheres use 4/3 x pi x r^3.
What is the difference between volume and capacity?
Volume is the space a shape takes up. Capacity is how much a container can hold inside, which is less once you subtract walls, fittings and any space you leave empty.
Why is the sphere formula 4/3 pi r cubed?
A sphere fills exactly two thirds of the cylinder that fits around it. That cylinder is 2 x pi x r^3, and two thirds of it is 4/3 x pi x r^3.
How do you find the volume of an irregular shape?
Split it into simple shapes, find each volume and add them up. For small solid objects, you can also sink them in water and measure how much the level rises.
References
These sources support the method or guidance used for Volume Calculator. Verify time-sensitive rules at the source.
Try the calculator
Open Volume Calculator, enter your scenario, and compare its supporting rows with this guide's method and checks.