Area, Volume, Surface Area, and Perimeter

Rectangle: The perimeter of a rectangle is the sum of the lengths of all four sides. Since opposite sides of a rectangle are equal in length, the formula simplifies to:

$\mathrm{Perimeter}=2\times \mathrm{length}+2\times \mathrm{width}$

Square: Since all sides of a square are equal in length, the perimeter of a square is four times the length of one side:

$\mathrm{Perimeter}=4\times \mathrm{side}$

Triangle: The perimeter of a triangle is the sum of the lengths of its three sides:

$\mathrm{Perimeter}=a+b+c$

where $a$, $b$, and $c$ are the lengths of the three sides.

Circle: The perimeter of a circle is called its circumference. The formula for the circumference of a circle is:

$\mathrm{Circumference}=2\pi r$

where $r$ is the radius of the circle.

Note: The formula for the circumference of a circle is provided on the SAT reference sheet.

Perimeter problems often involve finding the total distance around a shape or determining the dimensions of a shape given its perimeter.

Example: A rectangular garden has a perimeter of 60 feet. If the length of the garden is 20 feet, what is the width of the garden?

Using the formula for the perimeter of a rectangle:

$60=2\times 20+2\times \mathrm{width}$

Simplify: $60=40+2\times \mathrm{width}$

Subtract 40 from both sides: $20=2\times \mathrm{width}$

Divide both sides by 2: $\mathrm{width}=10$

Therefore, the width of the garden is 10 feet.

Rectangle: The area of a rectangle is the product of its length and width:

$\mathrm{Area}=\mathrm{length}\times \mathrm{width}$

Square: Since all sides of a square are equal in length, the area of a square is the square of the length of one side:

$\mathrm{Area}={\mathrm{side}}^2$

Triangle: The area of a triangle is half the product of its base and height:

$\mathrm{Area}=\frac{1}{2}\times \mathrm{base}\times \mathrm{height}$

Circle: The area of a circle is the product of π and the square of its radius:

$\mathrm{Area}=\pi r^2$

where $r$ is the radius of the circle.

Note: The formulas for the area of a rectangle, triangle, and circle are provided on the SAT reference sheet.

Area problems often involve finding the amount of space a shape occupies or determining the dimensions of a shape given its area.

Example: A rectangular room has an area of 120 square feet. If the length of the room is 12 feet, what is the width of the room?

Using the formula for the area of a rectangle:

$120=12\times \mathrm{width}$

Divide both sides by 12: $\mathrm{width}=10$

Therefore, the width of the room is 10 feet.

Rectangular Prism: The surface area of a rectangular prism is the sum of the areas of all six faces. Since opposite faces are equal in area, the formula simplifies to:

$\mathrm{Surface\; Area}=2\times \mathrm{length}\times \mathrm{width}+2\times \mathrm{length}\times \mathrm{height}+2\times \mathrm{width}\times \mathrm{height}$

Cube: Since all faces of a cube are equal in area, the surface area of a cube is six times the square of the length of one side:

$\mathrm{Surface\; Area}=6\times {\mathrm{side}}^2$

Triangular Prism: The surface area of a triangular prism is the sum of the areas of the two triangular bases and the three rectangular faces:

$\mathrm{Surface\; Area}=2\times \frac{1}{2}\times \mathrm{base}\times \mathrm{height}+\mathrm{perimeter}\times \mathrm{height}$

where $\mathrm{base}$ and $\mathrm{height}$ refer to the triangular base, and $\mathrm{perimeter}$ is the perimeter of the triangular base.

Note: The formulas for the surface area of a rectangular prism and cube are not typically provided on the SAT reference sheet, but they can be derived from the area formulas for rectangles and squares.

Surface area problems often involve finding the total area of all faces of a three-dimensional shape or determining the dimensions of a shape given its surface area.

Example: A cube has a surface area of 150 square inches. What is the length of one side of the cube?

Using the formula for the surface area of a cube:

$150=6\times {\mathrm{side}}^2$

Divide both sides by 6: ${\mathrm{side}}^2=25$

Take the square root of both sides: $\mathrm{side}=5$

Therefore, the length of one side of the cube is 5 inches.

Rectangular Prism: The volume of a rectangular prism is the product of its length, width, and height:

$\mathrm{Volume}=\mathrm{length}\times \mathrm{width}\times \mathrm{height}$

Cube: Since all sides of a cube are equal in length, the volume of a cube is the cube of the length of one side:

$\mathrm{Volume}={\mathrm{side}}^3$

Triangular Prism: The volume of a triangular prism is the product of the area of the triangular base and the height of the prism:

$\mathrm{Volume}=\frac{1}{2}\times \mathrm{base}\times \mathrm{height}\times \mathrm{height}$

where the first $\mathrm{base}$ and $\mathrm{height}$ refer to the triangular base, and the second $\mathrm{height}$ refers to the height of the prism.

Pyramid: The volume of a pyramid is one-third the product of the area of the base and the height of the pyramid:

$\mathrm{Volume}=\frac{1}{3}\times \mathrm{base\; area}\times \mathrm{height}$

Cylinder: The volume of a cylinder is the product of the area of the circular base and the height of the cylinder:

$\mathrm{Volume}=\pi r^{2}h$

where $r$ is the radius of the circular base, and $h$ is the height of the cylinder.

Cone: The volume of a cone is one-third the product of the area of the circular base and the height of the cone:

$\mathrm{Volume}=\frac{1}{3}\pi r^{2}h$

where $r$ is the radius of the circular base, and $h$ is the height of the cone.

Sphere: The volume of a sphere is four-thirds the product of π and the cube of its radius:

$\mathrm{Volume}=\frac{4}{3}\pi r^3$

where $r$ is the radius of the sphere.

Note: The formulas for the volume of a rectangular prism, cube, cylinder, cone, and sphere are provided on the SAT reference sheet.

Volume problems often involve finding the amount of space a three-dimensional shape occupies or determining the dimensions of a shape given its volume.

Example: A cylindrical container has a radius of 3 inches and a height of 8 inches. What is the volume of the container?

Using the formula for the volume of a cylinder:

$\mathrm{Volume}=\pi 3^2\times 8=\pi \times 9\times 8=72\pi$

Therefore, the volume of the container is $72\pi$ cubic inches.

When a shape is scaled (enlarged or reduced) by a factor, its linear dimensions (length, width, height, radius) are multiplied by that factor. The area, surface area, and volume scale differently:

• If linear dimensions are multiplied by a factor of $k$, then:
• The area and surface area are multiplied by a factor of $k^2$.
• The volume is multiplied by a factor of $k^3$.

Example: If the radius of a sphere is doubled, its surface area is multiplied by a factor of 4, and its volume is multiplied by a factor of 8.

A composite shape is a shape made up of two or more basic shapes. To find the area or volume of a composite shape, you can break it down into its basic components, find the area or volume of each component, and then add or subtract as necessary.

Example: A rectangular garden has a circular fountain in the center. The garden is 20 feet long and 15 feet wide, and the fountain has a radius of 3 feet. What is the area of the garden excluding the fountain?

The area of the garden is $20\times 15=300$ square feet.

The area of the fountain is $\pi 3^2=9\pi$ square feet.

Therefore, the area of the garden excluding the fountain is $300-9\pi$ square feet.

Density is the amount of mass per unit volume. The formula for density is:

$\mathrm{Density}=\frac{\mathrm{mass}}{\mathrm{volume}}$

Density problems often involve finding the mass of an object given its density and volume, or finding the volume of an object given its density and mass.

Example: A block of metal has a density of 7.8 grams per cubic centimeter and a volume of 50 cubic centimeters. What is the mass of the block?

Using the formula for density:

$7.8=\frac{\mathrm{mass}}{50}$

Multiply both sides by 50: $\mathrm{mass}=7.8\times 50=390$

Therefore, the mass of the block is 390 grams.