# Prisms

A prism is a **polyhedron** with two parallel, identical faces (called **bases**), and whose remaining faces (called **lateral faces**) are *parallelograms*.

The *lateral faces* are formed by joining corresponding vertices of the bases. The **altitude** (height) of a prism is the perpendicular distance between the bases. To find the volume of any prism, you use the following general formula (below).

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- Depending on the prism’s base, the “area of base” part will change.
- It’s best to use the word
*altitude*to designate the prism’s overall length because using the word*height*, for instance, could be confused with a base dimension, especially if it’s a triangular prism. Therefore, to prevent ambiguity,*altitude*is used.

**Rectangular Prism**

Let’s take a look at the** rectangular prism:**

Specific formulas for its **volume** and **surface area** (SA) are shown below. The volume formula can derived by replacing the well-known *area of a rectangle* formula (length × width) into where you see *area of base*. A rectangular prism is the only prism where any one of its faces can be considered the prism’s base. In addition, the SA formula is derived by adding up the area of each face. The formula below is currently in factored form, but it’s simply twice the base area **plus** twice the side areas plus twice the other sides areas.

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Let’s look at two examples where *volume* and *surface area* are calculated:

**Cubes**

If you’re dealing with a **cube ***− technically a square prism – *deriving its formula is relatively simple. Remember that a square is an equilateral quadrilateral, where all sides and angles are the same. Therefore, we can assign any letter for one side – the letter **a** has been chosen below:

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**Triangle Prism**

If the prism base were a **triangle**, the volume formula would change to the following:

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- Recall that the area of triangle is A = ½ × b × h.
**s**stands for**slant height**(the*slant height*is the hypotenuse of the right triangle formed by the height and half the base length – shown in the video below).

Lets review an example where these two features are calculated.

# Pyramids

Here’s the basic **anatomy of a pyramid:**

A summary of some important formulas related to pyramids are listed below. The term **lateral area** describes the surface area of all the faces **except** for the base. Therefore, if you’re asked to find the surface area, you first calculate the *lateral area*, and add it to the area of the base that you find individually. The **slant height** (s) – as opposed to the actual height of the solid – is also used to find the area of each individual face. Sometimes you’ll be forced to find the slant height using the Pythagorean theorem just to find the SA if it’s not provided.

On a similar note, the volume of any pyramid is always 1/3 the volume of its equivalent base prism. For example, if you have a square-based pyramid, its volume will be one third of a prism having the same base and altitude.

A **frustum **is a portion of a solid (usually a pyramid or a cone) which remains after its upper part has been cut off by a plane parallel to its base.

Let’s review an important problem pertaining to frustum pyramids.

# Cylinder, Cone, and Sphere

**Cylinder**

The volume and surface area formulas of a **cylinder** are written below:

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A *cylinder* is made up of two circle bases and a rectangle. Therefore, to find its SA, the area of its circular base is multiplied by 2 (parts 1 and 3 in the diagram), and the circumference acts as the width of its rectangular sleeve that you use to find its area, after which you sum the three parts.

**Cone**

A cone is a solid bounded by a plane region (its circular base) and the surface formed by line segments joining a point (the vertex) to points on the boundary of the base. If a perpendicular line can be extended from the center of its base and passes through the vertex, you have a **right circular cone **(left in diagram); otherwise it is called **oblique cone**. For the sake of simplicity, we’ll only focus on the former.

Just like with pyramids, cones are always 1/3 the volume of their equivalent cylinder having the same base and altitude.

The volume and lateral area formulas of a cone are provided below. Recall that **s** stands for slant height and is NOT equivalent to the actual height of the solid.

**Sphere**

A sphere is a three-dimensional version of a circle; think of it as a solid ball a marble. The two key formulas you need to know are shown below:

Let’s watch a sample problem where these two formulas are used and manipulated to find unknown values.