# Prisms

*Go to Surface Area or Volume.*

A prism is a solid object with:

- identical ends
- flat faces
- and the same
**cross section**all along its length !

A **cross section** is the shape made by cutting straight across an object.

The cross section of this object is a **triangle** ...

.. it has the same cross section all along its length ...

... so it's a **triangular prism.**

Then imagine it extending up from the sheet of paper ... |

## No Curves!

A prism is a polyhedron, which means all faces are flat!

No curved sides.

For example, **a cylinder is not a prism**, because it has curved sides.

## Bases

The ends of a prism are parallel

and each one is called a base.

## Sides

The side faces of a prism are parallelograms

(4-sided shapes with opposite sides parallel)

## These are all Prisms:

Square Prism: | Cross-Section: |

Cube: | Cross-Section: |

(yes, a cube is a prism, because it is a square all along its length) (Also see Rectangular Prisms ) | |

Triangular Prism: | Cross-Section: |

Pentagonal Prism: | Cross-Section: |

and more!

### Example: This hexagonal ice crystal.

It looks like a hexagon, but because it has some thickness it is actually a hexagonal prism!

*Photograph by NASA / Alexey Kljatov.*

## Regular vs Irregular Prisms

All the previous examples are **Regular** Prisms, because the cross section is regular (in other words it is a shape with equal edge lengths, and equal angles.)

Here is an example of an **Irregular Prism**:

Irregular Pentagonal Prism: | ||

Cross-Section | ||

It is "irregular" because the cross-section is not "regular" in shape. |

## Right vs Oblique Prism

When the two ends are perfectly aligned it is a Right Prism otherwise it is an Oblique Prism:

## Surface Area of a Prism

+ Base Perimeter × Length

### Example: What is the surface area of a prism where the base area is 25 m^{2}, the base perimeter is 24 m, and the length is 12 m:

^{2}+ 24 m × 12 m

^{2}+ 288 m

^{2}

**338 m**

^{2}

(Note: we have an Area Calculation Tool)

## Volume of a Prism

The Volume of a prism is the area of one end times the length of the prism.

Volume = Base Area × Length

### Example: What is the volume of a prism where the base area is 25 m^{2} and which is 12 m long:

^{2}× 12 m

**300 m**

^{3}Play with it here. The formula also works when it "leans over" (*oblique*) but remember that the height is at right angles to the base:

And this is why:

The stack can lean over, but still has the same volume

## More About The Side Faces

The side faces of a prism are parallelograms (4-sided shape with opposites sides parallel)

A prism can lean to one side, making it an **oblique prism**, but the two ends are still parallel, and the side faces are still parallelograms!

But if the two ends are **not parallel** it is **not a prism**.