Volume of Prisms

Volume measures the amount of 3D space a solid occupies. For any prism, the formula is: $V = A_\text{base} \times h$, where $A_\text{base}$ is the area of the cross-section and $h$ is the height (length) of the prism. The cross-section must be the face that stays constant as you move along the prism.

Cross-sections test a prism. Imagine slicing a solid at right angles to its length. A prism has the same shape and area at every parallel slice, so its uniform cross-section equals its base area.

A tapered roof block or pyramid is not a prism: its parallel slices change size. A combined solid is only a prism when its entire combined cross-section stays unchanged along its length.

For a rectangular prism (cuboid), the cross-section is a rectangle with area $A = l \times w$. Multiplying by the height $h$ gives:

$V = l \times w \times h$

This is also written $V = lwh$. Any order of multiplication works: $l \times w \times h = w \times l \times h$.

The cross-section is a triangle. Find its area first: $A_\triangle = \frac{1}{2} \times b \times h_\triangle$, where $b$ and $h_\triangle$ are the triangle's base and perpendicular height. Then multiply by the prism length $l$:

$V = \frac{1}{2} \times b \times h_\triangle \times l$

For an L-shaped cross-section, split it into two rectangles, find both areas, add them, then multiply by the prism length.

For a trapezium cross-section: $A = \frac{1}{2}(a+b)h$ where $a$ and $b$ are the parallel sides. Then $V = A \times l$.

Volume and capacity are related by:

• $1\ \text{cm}^3 = 1\ \text{mL}$
• $1\ 000\ \text{cm}^3 = 1\ \text{L}$
• $1\ \text{m}^3 = 1\ 000\ \text{L}$
• $1\ \text{m}^3 = 1\ 000\ 000\ \text{mL}$

To convert cm³ to mL: same number. To convert cm³ to L: divide by 1 000.