Volume of Prisms and Cylinders

A prism or cylinder has a constant cross-section along its length. Imagine stacking identical base shapes from one end to the other: volume is the base area multiplied by the distance between the bases. The base can be a rectangle, triangle, circle, or another polygon.

Use V = Bh

In V = Bh, B is an area, not a side length, and h is the perpendicular distance between the parallel bases. The formula works for right and oblique prisms and cylinders, provided h is measured perpendicular to the base. A slanted edge is not necessarily the height.

Worked examples

• Rectangular prism: base 4 cm by 3 cm, height 8 cm. B = 12 cm², so V = 12 × 8 = 96 cm³.
• Triangular prism: triangular base has base 6 m and perpendicular altitude 4 m; prism length is 10 m. B = ½ × 6 × 4 = 12 m², so V = 12 × 10 = 120 m³.
• Cylinder: radius 2 in and height 7 in. B = πr² = 4π in², so V = 4π × 7 = 28π in³, about 87.96 in³.

Watch the dimensions and units

If the base is a circle, use radius rather than diameter in πr². If the problem gives diameter d, first calculate r = d ÷ 2. The area contributes squared units and multiplying by a length creates cubic units: cm² × cm = cm³. Convert all measurements to a consistent unit before multiplying.

Common errors

• Using perimeter instead of base area.
• Using a slanted side as the perpendicular height.
• Forgetting the one-half in a triangular base area.
• Substituting diameter for radius in a cylinder’s base area.
• Giving square units or no units instead of cubic units.
• Rounding π too early when an exact answer in terms of π is accepted.

Find the base area before multiplying

For a prism, calculate the area of the polygonal base with its appropriate formula; for a cylinder, use πr². Then multiply that area by the perpendicular distance between the two congruent bases. A triangular prism needs one-half times the triangle's base times its perpendicular altitude, followed by multiplication by the prism length. A cylinder uses the circular radius, not its diameter, in the base-area formula.

Use the perpendicular height

The h in V = Bh measures the distance at a right angle between the parallel bases. In an oblique prism or cylinder, a slanted lateral edge may be longer than this height. Do not substitute the slant length merely because it appears as an outside edge in the diagram. If the diagram provides a perpendicular marker or an altitude, use that measurement.

Handle composite solids

A solid made from two nonoverlapping prisms can often be split into parts and the volumes added. For a cavity or cutout, subtract the removed volume from the larger solid. Make sure the parts do not overlap and that the dimensions refer to the correct base and height. A hollow cylinder, for example, uses the outer circular area minus the inner circular area, multiplied by the cylinder's perpendicular length.

Work backward and use scale

If volume and base area are known, height is V/B. If volume and height are known, base area is V/h; additional information may be needed to find a specific side or radius. If every linear dimension is multiplied by a scale factor k, volume is multiplied by k³. Doubling length, width, and height produces eight times the volume, not twice the volume.

Check dimensions and units

Area units multiply by a length unit to make cubic units: m² × m = m³. Convert all dimensions to a consistent unit before calculating. Estimate the base area and total volume to catch a misplaced decimal. Keep exact forms such as 28π cubic inches when requested, and round only once at the end when a decimal answer is needed.

Find prism volume from base area and height

A prism has a constant cross-section along its length. Its volume is the area of the base multiplied by the perpendicular height: V = Bh. For a rectangular prism, the base is a rectangle, so V = lwh. If a box has base 6 cm by 4 cm and height 10 cm, then V = (6×4)(10) = 240 cm³. The height is the distance between the bases, not a slanted side.

For a triangular prism, find the triangular base area first, B = ½bh, then multiply by the prism length. Keep the two different uses of h conceptually separate: the height of the triangle and the distance the triangle extends to form the prism. Label dimensions in a sketch to avoid multiplying the wrong values.

Use cylinder volume with radius and height

A cylinder’s volume is V = πr²h: area of the circular base times the perpendicular height. If a cylinder has radius 3 inches and height 8 inches, V = π(3²)(8) = 72π cubic inches, about 226.2 in³. If the problem gives diameter 6 inches, first divide by 2 to get r = 3. Squaring the diameter would make the answer four times too large.

A cylinder may be tilted, but the height used for the volume is still the perpendicular distance between its circular bases. A slanted side length does not replace h. Volume measures interior capacity, so use cubic units and decide whether an exact answer with π or a rounded decimal is requested.

Find a missing dimension and check units

If volume and base area are known, divide V by B to find prism height. If a rectangular prism has volume 240 cubic centimeters and base area 24 square centimeters, h = 240/24 = 10 centimeters. Units simplify from cm³/cm² to cm, which is a useful check. For a cylinder, solve V = πr²h for the requested variable only after identifying whether radius or height is unknown.

Convert linear measurements before calculating. A base measured in centimeters and a height in meters cannot be multiplied directly without conversion. If a container’s walls have thickness, the internal dimensions determine capacity while the external dimensions determine outer size. Use the dimensions the question identifies.

• For any prism, volume = base area × perpendicular height.
• For a cylinder, use πr²h and convert diameter to radius.
• Keep base height and prism height distinct for triangular prisms.
• Solve a missing dimension with inverse operations and check the unit.
• Use interior or exterior dimensions according to the question.

Exam takeaway

Calculate the base area, multiply by perpendicular height, and label the result in cubic units. For a cylinder, B = πr²; for a prism, use the area formula for its polygonal base.