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# Perimeter, area, and volume | Lesson

## What are perimeter, area, and volume?

The perimeter, start color #aa87ff, P, end color #aa87ff, of a polygon is the total length around the polygon's edges. When we add up all the side lengths of a polygon, we get its perimeter. For the figure below, the perimeter is:
start color #aa87ff, P, end color #aa87ff, equals, start color #11accd, s, start subscript, 1, end subscript, end color #11accd, plus, start color #1fab54, s, start subscript, 2, end subscript, end color #1fab54, plus, start color #ca337c, s, start subscript, 3, end subscript, end color #ca337c, plus, start color #a75a05, s, start subscript, 4, end subscript, end color #a75a05
The area, start color #e84d39, A, end color #e84d39, of a polygon is the amount of flat space inside the polygon's edges. Area is measured in square units such as square miles. Different polygons have their own area formulas.
The volume, start color #a75a05, V, end color #a75a05, of an object is the amount of space that object occupies. Volume is measured in cubic units such as cubic centimeters.

### What skills are tested?

• Calculating the perimeter of a polygon
• Calculating a side length given the perimeter
• Calculating the area of a polygon
• Calculating a length given the area
• Calculating the volume of an object
• Calculating a length given the volume

## Which perimeter formulas are on the test?

For a
square
, each side has the same length, s. The formula is:
start color #aa87ff, P, end color #aa87ff, equals, 4, s
For a
rectangle
, two sides have length start color #11accd, l, end color #11accd, and two sides have width start color #1fab54, w, end color #1fab54. The formula is:
start color #aa87ff, P, end color #aa87ff, equals, 2, left parenthesis, start color #11accd, l, end color #11accd, plus, start color #1fab54, w, end color #1fab54, right parenthesis
With each formula, we can also calculate the side lengths when start color #aa87ff, P, end color #aa87ff is given. To do so, we:
1. Write down the appropriate equation.
2. Plug in known values.
3. Solve for s, start color #11accd, l, end color #11accd, or start color #1fab54, w, end color #1fab54.

## Which area formulas are on the test?

For a
square
, each side has the same length, s. The formula is:
start color #e84d39, A, end color #e84d39, equals, s, squared
For a
rectangle
, two sides have length start color #11accd, l, end color #11accd, and two sides have width start color #1fab54, w, end color #1fab54. The formula is:
start color #e84d39, A, end color #e84d39, equals, start color #11accd, l, end color #11accd, start color #1fab54, w, end color #1fab54
For a
parallelogram
, we're concerned with the base start color #01a995, b, end color #01a995 and the height start color #ca337c, h, end color #ca337c. The formula is:
start color #e84d39, A, end color #e84d39, equals, start color #01a995, b, end color #01a995, start color #ca337c, h, end color #ca337c
For a
triangle
with base start color #01a995, b, end color #01a995 and the height start color #ca337c, h, end color #ca337c, the formula is:
start color #e84d39, A, end color #e84d39, equals, start fraction, 1, divided by, 2, end fraction, start color #01a995, b, end color #01a995, start color #ca337c, h, end color #ca337c
With each formula, we can also calculate one of the linear dimensions when start color #e84d39, A, end color #e84d39 is given. To do so, we:
1. Write down the appropriate equation.
2. Plug in known values.
3. Solve for the linear dimension.

## Which volume formulas are on the test?

For a
rectangular solid
with length start color #11accd, l, end color #11accd, width start color #1fab54, w, end color #1fab54, and height start color #ca337c, h, end color #ca337c, the formula is:
start color #a75a05, V, end color #a75a05, equals, start color #11accd, l, end color #11accd, start color #1fab54, w, end color #1fab54, start color #ca337c, h, end color #ca337c
We can also calculate one of the linear dimensions when start color #a75a05, V, end color #a75a05 is given. To do so, we:
1. Write down the volume equation.
2. Plug in known values.
3. Solve for start color #11accd, l, end color #11accd, start color #1fab54, w, end color #1fab54, or start color #ca337c, h, end color #ca337c.

TRY: CALCULATING PERIMETER
What is the perimeter of the rectangle above?

TRY: CALCULATING AREA
Triangle A, B, C is shown in the figure above. What is the area of triangle A, B, C ?

TRY: CALCULATING LENGTH FROM AREA
The base of a triangle is 6 inches long. If the area of the triangle is 12 square inches, what is the height of the triangle in inches?

TRY: CALCULATING VOLUME
The box for a board game is 11 inches long, 11 inches wide, and 3 inches tall. What is the volume of the box in cubic inches?
cubic inches

## Things to remember

ShapePerimeter (P) formula
Square
P, equals, 4, s
Rectangle
P, equals, 2, left parenthesis, l, plus, w, right parenthesis
ShapeArea (A) formula
Square
A, equals, s, squared
Rectangle
A, equals, l, w
Parallelogram
A, equals, b, h
Triangle
A, equals, start fraction, 1, divided by, 2, end fraction, b, h
ShapeVolume (V) formula
Rectangular solid
V, equals, l, w, h