SAT Math 12: Area and Volume

Area

Area measures the size of a two-dimensional region or planar surface. It is the two-dimensional analogue of length and has dimensions of $L^2$.

Surface area

Surface area is the area of the outer boundary of a three-dimensional object. Unlike volume, it is measured in squared units such as $\text{m}^2$.

SI unit of area

The SI unit of area is the square meter, $\text{m}^2$, which is the area of a square with side length $1\ \text{m}$.

Why are areas expressed in squared units?

Area is obtained by multiplying two perpendicular lengths, so its dimensions are $L \times L=L^2$. For example, length in meters produces area in square meters.

How do area conversion factors differ from length conversion factors?

The length conversion factor must be squared when converting areas. For example, because $1\ \text{cm}=10\ \text{mm}$, $1\ \text{cm}^2=100\ \text{mm}^2$.

How many square centimeters and square millimeters are in $1\ \text{m}^2$?

$1\ \text{m}^2=10{,}000\ \text{cm}^2=1{,}000{,}000\ \text{mm}^2$.

How many square meters are in $1\ \text{km}^2$?

$1\ \text{km}^2=1{,}000{,}000\ \text{m}^2$ because $1\ \text{km}=1000\ \text{m}$ and the conversion factor is squared.

Rectangle area

For a rectangle with length $l$ and width $w$, the area is $A=lw$.

Square area

For a square with side length $s$, the area is $A=s^2$.

Parallelogram area

The area of a parallelogram is $A=bh$, where $b$ is the base and $h$ is the perpendicular height, not the slanted side length.

Triangle area

The area of a triangle is $A=\frac{1}{2}bh$, where $b$ is a chosen base and $h$ is the perpendicular distance from that base to the opposite vertex.

Why is a triangle's area half the area of a parallelogram with the same base and height?

A diagonal divides such a parallelogram into two congruent triangles. Therefore, each triangle has area $\frac{1}{2}bh$.

Trapezoid area

For a trapezoid with parallel bases $a$ and $c$ and perpendicular height $h$, the area is $A=\frac{(a+c)h}{2}$.

Circle area

The area enclosed by a circle of radius $r$ is $A=\pi r^2$. In terms of diameter $d$, it is $A=\frac{\pi d^2}{4}$.

Cone surface area

For a cone with base radius $r$ and height $h$, total surface area is $A=\pi r(r+\ell)$, where the slant height is $\ell=\sqrt{r^2+h^2}$. This includes the base and lateral area.

Cylinder surface area

For a closed cylinder with radius $r$ and height $h$, the total surface area is $A=2\pi r(r+h)$: two circular bases plus the curved lateral surface.

Sphere surface area

The surface area of a sphere with radius $r$ is $A=4\pi r^2$, which is four times the area of a disk with the same radius.

Rectangular-prism surface area

For a rectangular prism with length $l$, width $w$, and height $h$, the surface area is $A=2(lw+lh+wh)$.

Prism surface area

For a prism with base area $B$, base perimeter $P$, and height $h$, the total surface area is $A=2B+Ph$.

Pyramid surface area

For a pyramid with base area $B$, base perimeter $P$, and slant height $L$, the total surface area is $A=B+\frac{1}{2}PL$.

Cube surface area

A cube with edge length $s$ has six congruent square faces, so its surface area is $A=6s^2$.

What is the volume of a prism?

The volume of a prism is $V=Bh$, where $B$ is the area of the base and $h$ is the perpendicular distance between the bases.

What is the volume of a cylinder?

A cylinder has volume $V=\pi r^2h$, where $r$ is the radius of its circular base and $h$ is its perpendicular height.

What is the volume of a pyramid?

A pyramid has volume $V=\frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the perpendicular height.

What is the volume of a cone?

A cone has volume $V=\frac{1}{3}\pi r^2h$, where $r$ is the radius of the circular base and $h$ is the perpendicular height.

What is the volume of a sphere?

A sphere with radius $r$ has volume $V=\frac{4}{3}\pi r^3$.

How do volume units differ from area units?

Volume is measured in cubic units because it multiplies three lengths, so its dimensions are $L^3$. For example, $1\ \text{m}^3=1{,}000{,}000\ \text{cm}^3$.