01 / 21

Volume With
Formulas

Master how to calculate the volume of rectangular prisms, triangular prisms, and cylinders using standard formulas and given dimensions.

SSAT Quantitative

What Is Volume?

Volume measures the amount of three-dimensional space a solid object occupies. Think of it as how much water you could pour inside a container, or how many tiny cubes you could stack to fill it up completely.

4 units 3 units 3 units

Why Does Volume Matter?

Volume shows up everywhere in real life: how much water a fish tank holds, how much soil fills a garden bed, or how much cereal fits in a box. On the SSAT, you will be asked to calculate volume using formulas and given dimensions.

Capacity
How much liquid a container holds (liters, gallons)
Packing
How many items fit inside a box or truck
Construction
How much concrete or material fills a space

Key Vocabulary

Base Area (\(B\))
The area of the flat bottom (or top) face of a 3D shape
Height (\(h\))
The perpendicular distance between the two bases (how tall)
Prism
A 3D shape with two identical, parallel bases connected by rectangles
Cylinder
A 3D shape with two identical, parallel circular bases

The Master Formula

For any prism or cylinder, volume follows one pattern:

Rectangular Prism Triangular Prism Cylinder

Rectangular Prism: \(V = l \times w \times h\)

length (l) width (w) height (h)

Triangular Prism: \(V = \frac{1}{2}bh_{\text{tri}} \times H\)

\(h_{\text{tri}}\) b (base) H (length)

Cylinder: \(V = \pi r^2 h\)

r h

Area vs. Volume

Area Volume
Dimensions 2D (flat surface) 3D (filled solid)
Measures Surface coverage Space inside
Units \(\text{cm}^2, \text{in}^2, \text{ft}^2\) \(\text{cm}^3, \text{in}^3, \text{ft}^3\)
Example Paint for a wall Water in a pool
Formula type \(l \times w\) \(l \times w \times h\)

Step-by-Step Method

1
Identify the shape. Is the solid a rectangular prism, triangular prism, or cylinder?
2
Pick the right formula. Write down the volume formula for that shape.
3
Plug in dimensions. Substitute the given values for length, width, height, radius, etc.
4
Calculate carefully. Multiply step by step. Watch for fractions and \(\pi\).
5
Label your answer. Always write cubic units (\(\text{cm}^3, \text{in}^3, \text{ft}^3\)).

Example 1: Rectangular Prism

A box has length 8 cm, width 5 cm, and height 3 cm. Find the volume.

8 cm 3 cm 5 cm

Example 2: Triangular Prism

A triangular prism has a triangle base of 6 in and triangle height 4 in. The prism is 10 in long. Find the volume.

4 in 6 in 10 in

Example 3: Cylinder

A cylinder has radius 7 cm and height 10 cm. Find the volume. Use \(\pi \approx 3.14\).

7 cm 10 cm

Example 4: Multi-Step Problem

A rectangular fish tank is 20 in long, 12 in wide, and 10 in tall. It is filled three-quarters full. How many cubic inches of water are in the tank?

1
Full volume: \(V = 20 \times 12 \times 10 = 2{,}400 \text{ in}^3\)
2
Three-quarters full: \(\frac{3}{4} \times 2{,}400 = \mathbf{1{,}800 \text{ in}^3}\)

Common Mistakes

Using Diameter for Radius
If given diameter \(d = 10\), the radius is \(r = 5\). Using 10 in \(\pi r^2\) gives 4 times the correct answer.
Forgetting the \(\frac{1}{2}\)
For a triangular prism, you must halve the base times height of the triangle. Skipping this doubles the volume.
Wrong Units
Volume uses cubic units (\(\text{cm}^3\)), not squared (\(\text{cm}^2\)). Writing \(\text{cm}^2\) means you calculated area, not volume.

Formula Summary

Shape Base Area (\(B\)) Volume Formula
Rectangular Prism \(l \times w\) \(V = l \times w \times h\)
Triangular Prism \(\frac{1}{2} b \, h_{\text{tri}}\) \(V = \frac{1}{2} b \, h_{\text{tri}} \times H\)
Cylinder \(\pi r^2\) \(V = \pi r^2 h\)
Cube \(s^2\) \(V = s^3\)

Question 1 of 5

A rectangular box has length 10 cm, width 4 cm, and height 6 cm. What is the volume?

Question 2 of 5

A triangular prism has a triangle with base 8 m and height 3 m. The prism is 12 m long. What is the volume?

Question 3 of 5

A cylinder has radius 5 in and height 8 in. What is the volume? Use \(\pi \approx 3.14\).

Question 4 of 5

A cube has side length 9 ft. What is its volume?

Question 5 of 5

A cylindrical can has diameter 12 cm and height 15 cm. What is the volume? Use \(\pi \approx 3.14\).