Surface Area and Volume of Prisms

A prism is a 3D shape with two identical, parallel bases connected by flat rectangular sides. The name of the prism comes from the shape of its base (e.g., a rectangular prism has rectangular bases, and a triangular prism has triangular bases).

When working with prisms, we often need to measure two things:

1. Volume: The amount of 3D space inside the prism.
2. Surface Area: The total area of all the outside faces that cover the prism.

Volume of a Prism

The volume of any prism is simply the area of its base multiplied by its height (or length).

$V = B \times h$

• $V$ = Volume
• $B$ = Area of the base
• $h$ = Height (or length) of the prism

Example: Triangular Prism

Question: A triangular prism has a base triangle with a base of $6$ and a height of $4$. The prism is $10$ units long. Find its volume.

1. Find the area of the base ($B$): The base is a triangle, so we use the triangle area formula $A = \frac{1}{2}bh$. $B = \frac{1}{2} \times 6 \times 4 = 12$
2. Multiply by the height of the prism ($h$): The prism is $10$ units long. $V = 12 \times 10 = 120$

The volume is $120$ cubic units.

Surface Area of a Prism

The surface area is the sum of the areas of all the faces of the prism. To find it, you calculate the area of each flat surface (the two bases and the rectangular sides) and add them all together.

For a rectangular prism, the formula is: $SA = 2(lw) + 2(lh) + 2(wh)$ where $l$ is length, $w$ is width, and $h$ is height.

Example: Rectangular Prism

Question: Find the volume and surface area of a rectangular prism with length $5$, width $3$, and height $4$.

Finding the Volume: $V = l \times w \times h$ $V = 5 \times 3 \times 4 = 60$ The volume is $60$ cubic units.

Finding the Surface Area: Calculate the area of all six faces:

• Top and Bottom: $2 \times (5 \times 3) = 2 \times 15 = 30$
• Front and Back: $2 \times (5 \times 4) = 2 \times 20 = 40$
• Left and Right: $2 \times (3 \times 4) = 2 \times 12 = 24$

Add them together: $SA = 30 + 40 + 24 = 94$

The surface area is $94$ square units.