Volume of Solids

Volume is the amount of three-dimensional space inside a solid object. If area measures how much flat surface something covers, volume measures how much it can hold.

Volume is always measured in cubic units — cubic inches (in$^3$), cubic feet (ft$^3$), cubic centimeters (cm$^3$), cubic meters (m$^3$), or cubic yards (yd$^3$).

Volume Formulas at a Glance

Rectangular Prism

A rectangular prism (a box shape) is the most common solid you’ll encounter. Multiply length times width times height.

$V = lwh$

Example 1: Find the volume of a box that is 8 ft long, 5 ft wide, and 3 ft tall.

$V = lwh = 8 \times 5 \times 3 = 120 \text{ ft}^3$

Answer: The volume is 120 cubic feet.

Example 2: A storage container measures 12 in by 8 in by 6 in. What is its volume?

$V = 12 \times 8 \times 6 = 576 \text{ in}^3$

Answer: The volume is 576 cubic inches.

Triangular Prism

A triangular prism has a triangular cross-section that extends along its length — think of a tent, a Toblerone box, or a roof ridge. The volume is the area of the triangular base times the length of the prism.

$V = \frac{1}{2}bh \times l$

where $b$ is the base of the triangle, $h$ is the height of the triangle, and $l$ is the length (depth) of the prism.

Example 3: A triangular prism has a triangular base with base 10 cm and height 7 cm, and the prism is 15 cm long. Find the volume.

Step 1 — Find the area of the triangular base:

$A = \frac{1}{2}bh = \frac{1}{2}(10)(7) = 35 \text{ cm}^2$

Step 2 — Multiply by the length:

$V = A \times l = 35 \times 15 = 525 \text{ cm}^3$

Answer: The volume is 525 cm$^3$.

Cylinder

A cylinder is a circular prism — think of a can, a pipe, or a round duct. The base is a circle with area $\pi r^2$, and you multiply by the height.

$V = \pi r^2 h$

Example 4: Find the volume of a cylinder with radius 4 cm and height 10 cm.

$V = \pi (4)^2 (10) = \pi (16)(10) = 160\pi \approx 502.65 \text{ cm}^3$

Answer: The volume is approximately 502.65 cm$^3$.

Cone

A cone has the same circular base as a cylinder, but it tapers to a point. Its volume is exactly one-third of the corresponding cylinder.

$V = \frac{1}{3}\pi r^2 h$

Example 5: A cone has a radius of 6 in and a height of 9 in. Find its volume.

$V = \frac{1}{3}\pi (6)^2 (9) = \frac{1}{3}\pi (36)(9) = \frac{324\pi}{3} = 108\pi \approx 339.29 \text{ in}^3$

Answer: The volume is approximately 339.29 in$^3$.

Pyramid (Square Base)

A pyramid with a square base has four triangular faces that meet at a single point (the apex). Like a cone, a pyramid’s volume is one-third of the prism with the same base and height.

$V = \frac{1}{3}Bh = \frac{1}{3}s^2 h$

where $s$ is the side length of the square base and $h$ is the perpendicular height from base to apex.

Example 6: A square pyramid has a base with sides of 8 m and a height of 12 m. Find the volume.

Step 1 — Find the area of the square base:

$B = s^2 = 8^2 = 64 \text{ m}^2$

Step 2 — Apply the pyramid volume formula:

$V = \frac{1}{3}Bh = \frac{1}{3}(64)(12) = \frac{768}{3} = 256 \text{ m}^3$

Answer: The volume is 256 m$^3$.

Pyramid (Triangular Base)

A pyramid with a triangular base — sometimes called a tetrahedron when all four faces are triangles — uses the same one-third principle. The base area is a triangle rather than a square.

$V = \frac{1}{3}Bh = \frac{1}{3} \times \frac{1}{2}b h_b \times h = \frac{1}{6}b \cdot h_b \cdot h$

where $b$ is the base of the triangular base, $h_b$ is the height of the triangular base, and $h$ is the pyramid height.

Example 7: A triangular pyramid has a base triangle with base 6 in and height 5 in, and the pyramid stands 9 in tall. Find the volume.

Step 1 — Find the area of the triangular base:

$B = \frac{1}{2}(6)(5) = 15 \text{ in}^2$

Step 2 — Apply the pyramid volume formula:

$V = \frac{1}{3}Bh = \frac{1}{3}(15)(9) = \frac{135}{3} = 45 \text{ in}^3$

Answer: The volume is 45 in$^3$.

Sphere

A sphere is a perfectly round 3D shape — a ball. Its volume depends only on the radius.

$V = \frac{4}{3}\pi r^3$

Example 8: Find the volume of a sphere with radius 3 ft.

$V = \frac{4}{3}\pi (3)^3 = \frac{4}{3}\pi (27) = 36\pi \approx 113.10 \text{ ft}^3$

Answer: The volume is approximately 113.10 ft$^3$.

Hemisphere

A hemisphere is exactly half of a sphere — think of a dome, a bowl, or a half-round planter. Since a full sphere is $\frac{4}{3}\pi r^3$, half of that gives us:

$V = \frac{2}{3}\pi r^3$

Example 9: A dome-shaped skylight has a radius of 2.5 ft. Find the volume of space enclosed by the dome.

$V = \frac{2}{3}\pi (2.5)^3 = \frac{2}{3}\pi (15.625) = \frac{31.25\pi}{3} \approx 32.72 \text{ ft}^3$

Answer: The volume is approximately 32.72 ft$^3$.

Real-World Application: Carpentry — Calculating Concrete for a Slab Pour

A carpenter needs to pour a concrete slab for a shed foundation. The slab will be 12 ft long, 10 ft wide, and 4 inches thick. Concrete is ordered in cubic yards. How much concrete should be ordered?

Step 1 — Convert all measurements to the same unit. Since the answer needs to be in cubic yards, convert everything to yards:

$12 \text{ ft} = 4 \text{ yd}, \quad 10 \text{ ft} = 3.333 \text{ yd}, \quad 4 \text{ in} = \frac{4}{36} \text{ yd} = 0.111 \text{ yd}$

Step 2 — Calculate the volume:

$V = 4 \times 3.333 \times 0.111 = 1.48 \text{ yd}^3$

Step 3 — Add 10% extra for waste and spillage:

$1.48 \times 1.10 = 1.63 \text{ yd}^3$

Answer: The carpenter should order approximately 1.63 cubic yards of concrete. In practice, you’d round up and order 2 cubic yards since concrete is typically sold in whole or half-yard increments.

An alternative approach is to work in feet first: $V = 12 \times 10 \times \frac{1}{3} = 40 \text{ ft}^3$, then convert using the fact that 1 yd$^3$ = 27 ft$^3$: $\frac{40}{27} \approx 1.48 \text{ yd}^3$.

Converting to Cubic Yards

Many real-world projects — concrete pours, gravel deliveries, mulch orders — require volume in cubic yards. The key conversion factor is:

$1 \text{ yd}^3 = 27 \text{ ft}^3$

Why 27? Because 1 yard = 3 feet, so 1 cubic yard = $3 \times 3 \times 3 = 27$ cubic feet.

To convert cubic feet to cubic yards, divide by 27:

$\text{yd}^3 = \frac{\text{ft}^3}{27}$

To convert cubic inches to cubic yards, divide by 46,656 (since $36^3 = 46{,}656$):

$\text{yd}^3 = \frac{\text{in}^3}{46{,}656}$

Example 10: A landscaper needs to fill a rectangular garden bed that is 18 ft long, 6 ft wide, and 8 inches deep with topsoil. Topsoil is sold by the cubic yard. How much should be ordered?

Step 1 — Convert the depth to feet:

$8 \text{ in} = \frac{8}{12} = \frac{2}{3} \text{ ft}$

Step 2 — Calculate the volume in cubic feet:

$V = 18 \times 6 \times \frac{2}{3} = 72 \text{ ft}^3$

Step 3 — Convert to cubic yards:

$V = \frac{72}{27} = 2.67 \text{ yd}^3$

Answer: The landscaper needs approximately 2.67 cubic yards of topsoil. Round up and order 3 cubic yards to account for settling and waste.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Volume measures three-dimensional space and is always in cubic units
• Rectangular prism: $V = lwh$ — multiply the three dimensions
• Triangular prism: $V = \frac{1}{2}bh \times l$ — area of the triangular base times the length
• Cylinder: $V = \pi r^2 h$ — the area of the circular base times the height
• Cone: $V = \frac{1}{3}\pi r^2 h$ — exactly one-third of the cylinder with the same base and height
• Pyramid: $V = \frac{1}{3}Bh$ — one-third of the base area times the height (works for any base shape)
• Sphere: $V = \frac{4}{3}\pi r^3$ — depends only on the radius
• Hemisphere: $V = \frac{2}{3}\pi r^3$ — exactly half of a sphere
• When working with real-world problems, make sure all measurements are in the same unit before calculating
• To convert cubic feet to cubic yards, divide by 27 (since $3^3 = 27$)

Return to Geometry for more topics in this section.