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Prisms are three-dimensions figures, also called polyhedrons. A polyhedron is a solid object with flat polygonal faces, straight edges, and sharp corners or vertices. A prism consists of two bases, which are congruent polygons, and additional faces that are parallelograms.

When the lateral faces of a prism are perpendicular to the bases, the prism is called a right prism. In a right prism, the lateral edges are perpendicular to both the bases and the base edges, forming right angles.

A prism is named for the shape of the base. For example, a triangular prism has a triangular base and three sides, like in the second image above. A rectangular prism has a rectangle for a base and four sides, a pentagonal prism has a five-sided base and five sides, a hexagonal prism has a six-sided base and six sides, and so on. Prisms are commonly used in geometry and are useful in many real-world applications, such as architecture, engineering, and design. They can be used to calculate the volume and surface area of various objects and structures. Understanding the properties and characteristics of prisms is important in higher-level mathematics and in many STEM fields.

The formula for the volume V of a prism is as follows:

$V=B\cdot h$

where $B$ is the area of the base and $h$ is the height of the prism.

**Example 1**

Find the volume of a prism with a height of 12 cm and with a right triangle for its base with legs of 6 cm and 8 cm.

Since the base is a right triangle, we replace B in our formula with the area of the base, which in a right triangle is $\frac{1}{2}\left({\mathrm{leg}}_{1}\right)\left({\mathrm{leg}}_{2}\right)$

$B=\frac{1}{2}\cdot \left(6\right)\cdot \left(8\right)$

$B=\frac{1}{2}\cdot \left(48\right)$

$=24{\mathrm{cm}}^{2}$

So the volume is

$V=Bh$

$=48\cdot 12$

$=576{\mathrm{cm}}^{3}$

For a rectangular prism, the volume formula is

$V=1wh$

where l equals the length of the base, w equals the width of the base, and h equals the height of the prism.

The formula for the surface area S of a prism is as follows:

$S=2B+Ph$

where B equals the area of the base of the prism, P equals the perimeter of the base of the prism, and h equals the height of the prism.

**Example 2**

Using the example prism from above and the Pythagorean theorem, the hypotenuse of the triangular base of the prism has a length of:

$\sqrt{{6}^{2}+{8}^{2}}$

$\sqrt{36+64}$

$\sqrt{100}$

$10\mathrm{cm}$

Therefore, the perimeter of the base of the prism is $6+8+10=24\mathrm{cm}$ . We already figured out above that the base area is $24{\mathrm{cm}}^{2}$ .

Therefore:

$S=\left(2B+Ph\right)$

$=2\left(24\right)+\left(24\right)\left(12\right)$

$=48+288$

$=336{\mathrm{cm}}^{2}$

If you ever need to find just the lateral surface area of a prism, that means the area of the sides without the bases. So the formula for that is simply $L=Ph$ .

Common Core: 6th Grade Math Flashcards

MAP 6th Grade Math Practice Tests

Learning about prisms can be a fun part of your student's geometry education. If they need a bit of help along the way, working with a math tutor is a great way to get it. A math tutor can make use of your student's learning style to bring the subject alive for them. Contact the Educational Directors at Varsity Tutors today to learn more about how tutoring can help your student learn about prisms and more.

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