Prism volume calculator

What is a prism?

A prism is a three-dimensional geometric shape with two parallel, congruent bases and rectangular lateral faces. The shape of the bases determines the type of prism. Prisms are known for their uniform cross-section along their entire length. Types of prisms include rectangular, triangular, and those with polygonal bases like pentagons or hexagons.

Types of prisms

1. Rectangular prism: Has bases shaped as rectangles.
2. Triangular prism: Bases are triangles.
3. Regular polygon base prism: Bases are regular polygons, such as hexagons or octagons.
4. Trapezoidal prism: Bases are trapezoids.

Formula

The volume of a prism can be calculated using a general formula. The key to calculating this volume is knowing the area of the prism’s base and its height.

$V = A \times l$

• $V$ is the volume.
• $A$ is the area of the base.
• $l$ is the length or height of the prism, which is the perpendicular distance between the two bases.

Rectangular prism

A rectangular prism has a straightforward volume formula because its base is a rectangle.

The formula is:

$V = l \times w \times h$

• $l$ is the length.
• $w$ is the width.
• $h$ is the height.

Triangular prism

For triangular prisms, the base is a triangle, and calculating its area requires different considerations based on the type of triangle.

Where $b$ is the base length of the triangle, and $h_{\text{base}}$ is the height of the triangle.

Prisms with polygonal bases

For prisms with regular polygon bases, the area can be calculated using the formula for a regular polygon:

• $n$ is the number of sides.
• $s$ is the side length.

Trapezoidal prism

A prism with a trapezoidal base has its base area calculated by:

• $a$ and $b$ are the lengths of the parallel sides.
• $h_{\text{trap}}$ is the height of the trapezoid.

Examples

Rectangular prism example

Consider a rectangular prism with a length of 10 cm, a width of 4 cm, and a height of 5 cm. The volume is:

$V = 10 \times 4 \times 5 = 200 \, \text{cm}^3$

Triangular prism example

For a triangular prism with base length 6 cm, base height 3 cm, and prism height 10 cm:

$A_{\text{triangle}} = \frac{1}{2} \times 6 \times 3 = 9 \, \text{cm}^2$ $V = 9 \times 10 = 90 \, \text{cm}^3$

Regular hexagonal prism example

If you have a hexagonal base with a side length of 2 cm and prism height 10 cm:

$A_{\text{hexagon}} = \frac{6 \times 2^2}{4 \times \tan\left(\frac{\pi}{6}\right)} \approx 10.39 \, \text{cm}^2$ $V \approx 10.39 \times 10 = 103.9 \, \text{cm}^3$

Trapezoidal prism example

Given a trapezoidal base with parallel side lengths of 5 cm and 7 cm, a height of 4 cm, and prism height of 12 cm:

$A_{\text{trapezoid}} = \frac{1}{2} \times (5 + 7) \times 4 = 24 \, \text{cm}^2$ $V = 24 \times 12 = 288 \, \text{cm}^3$

Frequently Asked Questions

How to calculate the prism volume if the base is a pentagon?

For a pentagonal base, calculate the area using:

$A_{\text{pentagon}} = \frac{5 \times s^2}{4 \times \tan\left(\frac{\pi}{5}\right)}$

Then multiply by the prism length $l$.

What is the prism volume if the base is a circle?

Note that a prism with circular base is a cylinder. The formula to find the volume is:

$V = \pi \times r^2 \times h$

More information about the volume of a cylinder can be found in cylinder volume calculator

How many different prisms can exist based on their base shapes?

Theoretically, an infinite number of prisms can exist if you consider any polygonal shape for the base. The most common are triangular, rectangular, pentagonal, and hexagonal prisms.

How is volume affected by doubling the prism height?

Doubling the height of the prism doubles its volume because volume depends linearly on height ($V = A \times l$).

Are prisms always symmetrical?

While prisms have congruent bases and identical lateral faces in terms of symmetry between bases, the lateral faces may not be symmetrical when considering other axes depending on the shape of the base.