Volume formulas for common shapes

Volume is the amount of three-dimensional space a shape occupies. The core idea is surprisingly simple: take the area of the base and multiply by the height. That “height” is the direction that extends the shape into 3D. Many formulas you memorize in school are just special cases of this rule.

A rectangular prism is the easiest: it’s like stacking layers of rectangles. If the base rectangle is length × width, then stacking up to height H gives: V = L × W × H. This is also why volume units are cubic—you're multiplying three lengths.

A cube is a special rectangular prism where every side is equal. If the side length is s, then V = s³. The “³” tells you it’s length multiplied by itself three times. A 4 cm cube has 4 × 4 × 4 = 64 cm³ of volume.

A cylinder is like a stack of circles. The base area is a circle: A = πr². Multiply by height h: V = πr²h. Cylinders show up everywhere: water bottles, pipes, cans, and tanks.

A cone is “one-third of a cylinder” with the same base and height (imagine a cylinder filled with three identical cones). So you take the cylinder volume and multiply by 1/3: V = (1/3)πr²h. This is useful for funnels, ice cream cones, and pile-like shapes.

Spheres are the classic “ball” shape. The formula is: V = (4/3)πr³. Notice the radius is cubed because the sphere grows in all directions as you increase size. Spheres are common in physics, chemistry, sports, and 3D printing.

A triangular prism is a prism with a triangle base. Start with triangle area: A = (1/2) × b × h (base times triangle height, divided by 2), then multiply by prism length L: V = (1/2) b h L. This appears in roof shapes, ramps, and many engineering cross-sections.

One viral study trick: treat volume formulas as “area × length.” It turns memorization into logic. If you know the area formula for the base, you can rebuild the volume formula.

Step-by-step volume examples

A storage box measures 40 cm (length) × 30 cm (width) × 25 cm (height). Multiply the three numbers: 40 × 30 × 25 = 30,000 cm³. Because 1 L = 1,000 cm³, that’s 30,000 ÷ 1,000 = 30 liters. So the box holds about 30 L (ignoring wall thickness).

A water bottle has radius 3.5 cm and height 22 cm. Use V = πr²h: r² = 3.5² = 12.25. Multiply: 12.25 × 22 = 269.5. Now multiply by π: 269.5π ≈ 846.7 cm³. Convert to liters: 846.7 ÷ 1,000 ≈ 0.847 L (about 847 mL).

A basketball has radius about 12 cm (rough estimate). V = (4/3)πr³: r³ = 12³ = 1,728. Multiply: (4/3) × 1,728 = 2,304. Then 2,304π ≈ 7,238 cm³. That’s roughly 7.24 liters of air inside the ball.

If you want to sanity-check answers quickly: compare to a familiar unit. A standard soda can is ~355 mL. A 1-liter bottle is 1,000 mL. If your “bottle” calculation gives 20 liters, something is likely wrong with a unit or a radius/diameter mix-up.

What the calculator is doing (in plain English)

This tool follows a consistent workflow so you can trust the result:

• 1) You choose a shape. That determines which formula is used.
• 2) You enter dimensions. We validate that numbers are positive and present.
• 3) We compute volume in the input unit. For example, if you enter centimeters, the raw result is in cm³.
• 4) We convert to common units. We show liters and m³ (plus ft³/in³ when it’s helpful).
• 5) We show the formula you used. So you can copy it into homework or a report.

The only “gotcha” in volume is unit scale. Switching from cm to m isn’t a 100× change in the final volume—it’s 1,000,000×, because you cube the conversion. That’s why a tiny unit mistake makes the answer look wildly wrong.

Cylinders, cones, and spheres use radius (half the diameter). If you only know the diameter, divide by 2 first. Using diameter directly in a radius formula makes the answer 4× too large for cylinders/cones (because r²), and 8× too large for spheres (because r³). That’s a huge error.

Frequently Asked Questions

• What is volume measured in?

  Volume uses cubic units (cm³, m³, in³, ft³) because it’s “length × length × length.” In daily life you’ll often see liters (L) and milliliters (mL) for capacity: 1 L = 1,000 cm³ and 1 mL = 1 cm³.

• What if my shape isn’t listed?

  Break it into pieces you can model (boxes, cylinders, cones), compute each part, then add or subtract. This “composite volume” approach is how engineers estimate tanks, packaging, and 3D printed designs.

• Do I need exact π?

  For most real-world use, π ≈ 3.14159 is plenty. In schoolwork, you may be asked to keep answers in “π form” (like 269.5π cm³). This calculator outputs a decimal, but you can still write the symbolic form using the shown formula.

• Why does changing units change the number so much?

  Because volume is cubic. If 1 m = 100 cm, then 1 m³ = (100 cm)³ = 1,000,000 cm³. That’s why unit mistakes blow up.

• Is “capacity” the same as volume?

  They’re closely related. “Volume” is the geometric space occupied by the shape, while “capacity” often means how much liquid a container can hold (which is the interior volume). Thickness and rounded edges can reduce capacity.

• Can volume ever be negative?

  In geometry, no. If you see a negative value, it’s a sign of an input error or a formula being used incorrectly.