Table of Contents
Understanding Volume
In the parent chapter “Area and Volume,” you met the general ideas of measuring two-dimensional area and three-dimensional volume. Here we focus specifically on volume of solids: how to measure “how much space” a solid object takes up, and the basic formulas used in school geometry.
When we talk about volume here, we always mean three-dimensional volume measured in cubic units such as $cm^3$, $m^3$, or $in^3$.
A useful mental picture: if area can be thought of as “how many unit squares cover a surface,” then volume can be thought of as “how many unit cubes fill a space.”
Many solids you meet in school geometry are prisms, pyramids, cylinders, cones, and spheres. Each has its own volume formula, but they are closely related and often built around the idea:
$$
\text{volume} = \text{area of base} \times \text{height}
$$
or a simple fraction of that.
We will go through the main solid types one by one.
Volume of Prisms
A prism is a solid that has:
- Two parallel, congruent faces (the bases), and
- Side faces that are parallelograms (often rectangles in simple examples).
If you know the area of the base and the height of the prism (the distance between the two bases), the volume is:
$$
V_{\text{prism}} = (\text{area of base}) \times (\text{height})
$$
This works for any prism, no matter what shape the base is (triangle, rectangle, hexagon, etc.).
Rectangular Prism (Box)
A rectangular prism (a box) has a rectangular base and rectangular sides. If its three edge lengths are $l$ (length), $w$ (width), and $h$ (height):
- Area of base $= l \times w$
- Height $= h$
So:
$$
V_{\text{rectangular prism}} = lwh
$$
This is one of the most commonly used volume formulas.
Cube
A cube is a special rectangular prism where all edges are the same length $s$:
- $l = w = h = s$
So:
$$
V_{\text{cube}} = s^3
$$
You can think of this as counting how many $1 \times 1 \times 1$ cubes fit along length, width, and height, then multiplying.
Other Prisms
If a prism has a base that is:
- a triangle, use the triangle’s area for the base;
- a regular polygon, use its area formula (covered in the polygons and area chapters), then multiply by the height.
In every case:
$$
V = (\text{area of base}) \times h
$$
Volume of Cylinders
A cylinder is like a prism with circular bases. It has:
- Two parallel circular bases of radius $r$,
- Height $h$ (distance between the bases).
Area of a circular base is:
$$
A_{\text{base}} = \pi r^2
$$
So the volume of a cylinder is:
$$
V_{\text{cylinder}} = \pi r^2 h
$$
This matches the prism pattern: “area of base times height,” where the base is a circle.
Volume of Pyramids
A pyramid has:
- One base (any polygon),
- Triangular faces meeting at a single point (the apex).
For a pyramid with base area $B$ and height $h$ (distance from base to apex, measured perpendicularly):
$$
V_{\text{pyramid}} = \frac{1}{3} B h
$$
So a pyramid’s volume is one-third of the volume of a prism with the same base and height.
You do not need to prove this here; just remember the pattern “one-third base times height.”
Right Pyramids and Slant Height
A right pyramid is one where the apex is directly “above” the center of the base. In problems, you might see:
- Height $h$: the perpendicular distance from the apex to the base.
- Slant height: the length along a triangular face from the apex to an edge of the base.
Be careful: volume always uses the perpendicular height, not the slant height. If only the slant height is given, you often need other information and the Pythagorean theorem (from the triangles chapter) to find the true height $h$.
Volume of Cones
A cone is like a pyramid, but with a circular base:
- Base: circle of radius $r$,
- Height $h$: perpendicular distance from the apex to the center of the base.
Since a cone is the circular version of a pyramid, its volume formula mirrors that of a pyramid:
$$
V_{\text{cone}} = \frac{1}{3} (\text{area of base}) \times h
= \frac{1}{3} \pi r^2 h
$$
Again, watch the difference between:
- Height $h$ (used in the formula, perpendicular),
- Slant height (length along the side of the cone).
The formula uses the perpendicular height.
Volume of Spheres
A sphere is the set of all points in space at a fixed distance (the radius $r$) from a center point. Common examples are balls and bubbles.
The volume formula for a sphere is:
$$
V_{\text{sphere}} = \frac{4}{3} \pi r^3
$$
This is one of the standard formulas you should know. Notice the pattern of a constant factor times $r^3$ (cubic in the radius).
Composite Solids
Many real-world shapes are composite solids: combinations of basic solids (prisms, cylinders, cones, pyramids, spheres).
The general strategy:
- Break the shape into simpler pieces whose volumes you know how to find.
- Compute each simple volume.
- Combine them appropriately:
- If shapes are added together (stacked, glued), add their volumes.
- If one shape is removed from another (like a hole), subtract the missing volume.
Examples of reasoning:
- A pillar with a cylindrical base and a rectangular block on top:
- Find cylinder volume using $V = \pi r^2 h$,
- Find block volume using $V = lwh$,
- Add them.
- A cylindrical container with a conical “funnel” or tip:
- Find cone volume and cylinder volume,
- Add or subtract depending on the shape.
When composite solids are involved, understanding which dimensions belong to which part is more important than remembering many new formulas.
Units and Conversions for Volume
Volume uses cubic units. That means:
- If length is measured in cm, volume is in $cm^3$.
- If length is measured in m, volume is in $m^3$.
Key idea: cubes scale by powers of 3 in linear unit conversions. For example:
- $1\text{ m} = 100\text{ cm}$
- Therefore:
$$
1\text{ m}^3 = (100\text{ cm})^3 = 1{,}000{,}000\text{ cm}^3
$$
This grows very quickly. Always convert lengths first and then apply the volume formula.
Using Volume in Word Problems
Volume formulas are often used in practical questions, such as:
- How much liquid a container can hold.
- How much material is needed to build something.
- How much space an object occupies.
Typical patterns:
- Given dimensions → find volume: plug into the appropriate formula.
- Given volume → find a dimension: use the formula and solve for the unknown using basic algebra.
Examples of rearranging:
- For a rectangular prism with volume $V$ and base area $B$:
$$
V = Bh \quad \Rightarrow \quad h = \frac{V}{B}
$$ - For a cylinder:
$$
V = \pi r^2 h \quad \Rightarrow \quad h = \frac{V}{\pi r^2}
$$
The formulas themselves come from geometry; solving them for different variables uses algebra skills from earlier chapters.
Summary of Key Volume Formulas
For a solid with height $h$ and base area $B$:
- Prism:
$$
V = Bh
$$ - Rectangular prism (box):
$$
V = lwh
$$ - Cube:
$$
V = s^3
$$ - Cylinder:
$$
V = \pi r^2 h
$$ - Pyramid:
$$
V = \frac{1}{3}Bh
$$ - Cone:
$$
V = \frac{1}{3}\pi r^2 h
$$ - Sphere:
$$
V = \frac{4}{3}\pi r^3
$$
These are the standard school-level formulas for the volume of common solids. Understanding which one applies, what each symbol means, and how to handle units is the core of this chapter.