Area and Volume

Understanding Area and Volume

Area and volume are two ways of measuring the “size” of geometric objects, but they apply to different kinds of shapes.

Area measures how much flat surface a shape covers. It is used for 2‑dimensional (2D) shapes like rectangles, triangles, and circles.
Volume measures how much space a solid object occupies. It is used for 3‑dimensional (3D) shapes like cubes, prisms, cylinders, and spheres.

They are related but not the same: a square and a cube can have sides of the same length, but one has only area, the other has volume.

Units:

Area is measured in square units (such as $\text{cm}^2$, $\text{m}^2$).
Volume is measured in cubic units (such as $\text{cm}^3$, $\text{m}^3$).

You should already be familiar with basic shapes (points, lines, angles, triangles, quadrilaterals, circles, solids) from earlier geometry chapters. Here we focus on how to compute and compare their areas and volumes.

Area of Basic Plane Figures

In this section, we consider standard formulas. You do not need to memorize how they are derived yet; focus on what they are and how to use them.

Area of Rectangles and Squares

A rectangle has length $l$ and width $w$ (sometimes called base and height). Its area is
$$
A_{\text{rectangle}} = l \cdot w.
$$

A square is a special rectangle with all sides equal, side length $s$:
$$
A_{\text{square}} = s^2.
$$

If dimensions are in meters, area is in square meters, $\text{m}^2$.

Area of Parallelograms

A parallelogram has opposite sides parallel, with base $b$ and height $h$ (height is the perpendicular distance between the two bases):
$$
A_{\text{parallelogram}} = b \cdot h.
$$

Notice that “slant” does not change the area; only base and perpendicular height matter.

Area of Triangles

A triangle can be thought of as half a parallelogram with the same base and height. If $b$ is a base and $h$ is the corresponding height (altitude):

$$
A_{\text{triangle}} = \frac{1}{2} b h.
$$

No matter which side you choose as the base, you must use the height that is perpendicular to that side.

Area of Trapezoids

A trapezoid has exactly one pair of parallel sides, called bases $b_1$ and $b_2$. Let $h$ be the perpendicular distance between them:
$$
A_{\text{trapezoid}} = \frac{1}{2} (b_1 + b_2) h.
$$

This formula can be viewed as the height times the average of the two bases.

Area of Circles

A circle with radius $r$ has area
$$
A_{\text{circle}} = \pi r^2.
$$

Here $\pi$ (pi) is a constant approximately equal to $3.14159$.

If you know the diameter $d$ instead, use $r = \dfrac{d}{2}$:
$$
A_{\text{circle}} = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}.
$$

Area of Composite and Irregular Shapes

Many shapes in practical problems are not simple rectangles or circles. A common strategy is to break complex shapes into simpler pieces whose areas you know how to find.

Adding Areas of Parts

If a region is split into non-overlapping simple shapes, its area is the sum of the areas of the parts.

Example structure:

Divide the shape into rectangles, triangles, etc.
Compute each area.
Add them:
$$
A_{\text{total}} = A_1 + A_2 + A_3 + \dots.
$$

Subtracting Areas (Cut‑Outs)

If a region is like a “big shape” with one or more smaller shapes removed (for example, a garden with a circular pond inside a rectangular lot), then:
$$
A_{\text{remaining}} = A_{\text{big}} - A_{\text{cut\,out}}.
$$

You can also combine these methods:

Break into several parts,
Add some areas,
Subtract others that are removed.

Approximation of Irregular Areas

For very irregular shapes where exact formulas are not obvious, you can approximate:

Cover the shape with a grid of squares and estimate how many squares fit inside.
Use rectangles or trapezoids whose total area is close to the shape.

These approximation ideas are the beginning of what later leads to integral calculus, but at this level we just use them for rough estimates.

Surface Area of Solids

While area usually refers to flat 2D figures, for 3D solids we often need the surface area: the total area of all the faces (sides) of the solid.

Surface area is still measured in square units.

Nets of Solids

A useful tool is the net of a solid: a 2D pattern that folds up into the 3D shape. Each face in the net is a 2D shape whose area can be calculated.

Process:

Imagine unfolding the solid into a flat net.
Identify each face (rectangle, square, circle, etc.).
Compute the area of each face.
Add them to get total surface area.

Surface Area of Prisms

A prism has two identical parallel faces (bases) and rectangular side faces.

Let each base have area $B$.
Let the perimeter of the base be $P$.
Let the height (length) of the prism be $h$.

Then:

Total area of the two bases: $2B$.
Area of the side faces (lateral area): $P \cdot h$.

So the total surface area is
$$
S_{\text{prism}} = 2B + P h.
$$

This formula applies to any prism (rectangular, triangular, etc.) as long as you know $B$ and $P$ of the base.

Surface Area of Cylinders

A cylinder is like a prism with circular bases.

Radius of base: $r$.
Height: $h$.

Pieces:

Two circular bases: each area $\pi r^2$, so total $2 \pi r^2$.
Side surface (lateral area): if you “unroll” it, it becomes a rectangle with height $h$ and width equal to the circumference of the circle, $2 \pi r$. So lateral area is
$$
(2 \pi r) \cdot h = 2 \pi r h.
$$

Total surface area:
$$
S_{\text{cylinder}} = 2 \pi r^2 + 2 \pi r h.
$$

Surface Area of Pyramids and Cones

A pyramid has a polygon base and triangular side faces meeting at a point (the apex).

A cone has a circular base and a curved surface meeting at a point.

These solids have:

Area of the base (a polygon or circle),
Plus the total area of the side faces (called the lateral area).

For many problems at this level, you:

Compute the area of the base.
Compute the area of each triangular side face (for pyramids) or use a given formula (for cones).
Add them to get the total surface area.

Volume of Solids

Volume measures how much 3D space a solid occupies. It is always in cubic units.

At this level, you mainly work with standard formulas and the idea that stretching a solid in one direction multiplies its volume by the same factor.

Volume of Prisms

A prism has a constant cross-section when you slice it perpendicular to its height. The key idea:

Let $B$ = area of the base.
Let $h$ = height (distance between the two bases).

Then the volume is
$$
V_{\text{prism}} = B h.
$$

This works for any prism, no matter the shape of the base:

For a rectangular prism (box) with length $l$, width $w$, and height $h$:
$$
B = l w,\quad
V = l w h.
$$
For a triangular prism with base triangle area $B$ and height $h$, same formula $V = B h$.

Volume of Cylinders

A cylinder is a prism with a circular base.

Base radius: $r$.
Height: $h$.

Base area is $B = \pi r^2$, so
$$
V_{\text{cylinder}} = \pi r^2 h.
$$

This matches the idea “volume = area of base $\times$ height.”

Volume of Pyramids

A pyramid has a base of area $B$ and height $h$ (height is measured from the apex straight down to the base).

The volume formula is:
$$
V_{\text{pyramid}} = \frac{1}{3} B h.
$$

This is true for any base shape (triangular, square, etc.). Compared to a prism with the same base and height, the pyramid’s volume is exactly one-third.

Volume of Cones

A cone is like a pyramid with a circular base.

Base radius: $r$.
Height: $h$.

Base area: $B = \pi r^2$, so
$$
V_{\text{cone}} = \frac{1}{3} \pi r^2 h.
$$

Again, this is one-third the volume of a cylinder with the same base and height.

Volume of Spheres

A sphere is the set of all points at a fixed distance (the radius) from a center.

Radius: $r$.

The volume of a sphere is
$$
V_{\text{sphere}} = \frac{4}{3} \pi r^3.
$$

Comparing and Scaling Areas and Volumes

Area and volume react differently when you change the size of a shape.

Scaling Lengths and Area

If every length in a 2D shape is multiplied by a factor $k$ (for example, all sides doubled, $k=2$):

The new area is multiplied by $k^2$.

So:

Double all side lengths ($k=2$) $\Rightarrow$ area becomes $4$ times larger.
Triple all side lengths ($k=3$) $\Rightarrow$ area becomes $9$ times larger.

Scaling Lengths and Volume

If every length in a 3D solid is multiplied by a factor $k$:

The new volume is multiplied by $k^3$.

So:

Double all lengths ($k=2$) $\Rightarrow$ volume becomes $8$ times larger.
Triple all lengths ($k=3$) $\Rightarrow$ volume becomes $27$ times larger.

This explains why small changes in size can lead to large changes in volume.

Using Units and Conversions

Because area and volume use squared and cubed units, unit conversions behave differently from simple length conversions.

Area Unit Conversions

If
$$
1 \text{ m} = 100 \text{ cm},
$$
then
$$
1 \text{ m}^2 = (100 \text{ cm})^2 = 10{,}000 \text{ cm}^2.
$$

In general, if $1$ large unit $= k$ small units, then
$$
1 \text{ (large unit)}^2 = k^2 \text{ (small units)}^2.
$$

Volume Unit Conversions

Similarly, with volume:
$$
1 \text{ m}^3 = (100 \text{ cm})^3 = 1{,}000{,}000 \text{ cm}^3.
$$

In general, if $1$ large unit $= k$ small units, then
$$
1 \text{ (large unit)}^3 = k^3 \text{ (small units)}^3.
$$

When solving problems, always check:

Are all your lengths in the same unit?
Are you reporting the answer in the requested unit (e.g., $\text{cm}^2$, $\text{m}^3$)?

Applications of Area and Volume

Area and volume appear frequently in practical situations:

Area:

Flooring or painting a room (how many tiles, how much paint).
Size of land or fields.
Cross-sectional area in physics or engineering (e.g., area of a pipe opening).

Volume:

Capacity of containers (bottles, tanks, pools).
Amount of material needed to build a solid object.
Displacement of fluids (how much water a solid object pushes aside).

Most real problems require:

Identifying the relevant shape(s),
Choosing or recalling the correct formula(s),
Carefully substituting the given measurements,
Using appropriate units and conversions.

With these tools, you can analyze and compare both the flat space that shapes cover and the solid space that objects occupy.