Perimeter Of Rectangle

What Is the Perimeter of a Rectangle?

The perimeter of a rectangle is the total distance around its outer boundary. Imagine you are walking along the edge of a rectangular park; the total distance you walk to get back to your starting point is the park's perimeter. It's a one-dimensional measurement of length.

To understand perimeter, we first need to remember the key properties of a rectangle:

• It is a four-sided shape (a quadrilateral).
• All four of its interior angles are right angles (${90}^{\circ }$).
• Its opposite sides are equal in length and parallel to each other.

We typically label the two longer sides as the length (often denoted by $l$) and the two shorter sides as the width (often denoted by $w$). Because opposite sides are equal, we know that a rectangle has two sides of length $l$ and two sides of length $w$.

To find the perimeter, we simply add the lengths of all four sides together. If you start at one corner and go all the way around, you would travel a distance of one length, then one width, then another length, and finally another width. This gives us our most basic, foundational understanding of the perimeter calculation:

Perimeter = length + width + length + width

Using our variables, we can write this as:

This simple idea of adding up all the side lengths is the core concept behind perimeter. From this starting point, we can derive more streamlined formulas that make the calculation faster and easier.

What Are the Formulas for Calculating a Rectangle's Perimeter?

While adding all four sides always works, mathematicians prefer to simplify expressions to make them more efficient. Starting from $P=l+w+l+w$, we can group the like terms together. We have two lengths ($l+l$) and two widths ($w+w$). This leads us to our first official formula.

Formula 1: The Sum of Doubled Sides

By combining the identical sides, we get two lengths ($2l$) and two widths ($2w$). Adding these together gives us the perimeter.

This formula tells us to double the length, double the width, and then add those two results together. It's a very direct and easy-to-understand method.

Formula 2: Doubling the Sum of Sides

Another way to look at the expression $2l+2w$ is to use the distributive property of multiplication. We can see that the number $2$ is a common factor in both terms. By factoring out the $2$, we get our second common formula.

This formula tells us to first add the length and width together, and then multiply that sum by $2$. This is often quicker for mental math, as you only have to do one addition and one multiplication.

Which formula should you use? It doesn't matter! Both formulas are mathematically equivalent and will always give you the exact same answer. You can choose whichever one you find more intuitive or easier to remember. Let's prove they are the same with a quick example. If $l=5$ and $w=3$:

• Using Formula 1: $P=2(5)+2(3)=10+6=16$
• Using Formula 2: $P=2(5+3)=2(8)=16$

As you can see, the result is identical. Pick the one that works best for you!

How Do You Calculate the Perimeter of a Rectangle?

Calculating the perimeter is a straightforward process. By following these steps, you can ensure you get the correct answer every time.

1. Identify the Knowns: Read the problem carefully and identify the given length ($l$) and width ($w$) of the rectangle.
2. Check the Units: Make sure the length and width are in the same unit of measurement (e.g., both are in inches, or both are in centimeters). If they are not, you must convert one of them before proceeding.
3. Choose Your Formula: Select one of the perimeter formulas: $P=2l+2w$ or $P=2(l+w)$.
4. Substitute the Values: Replace the variables $l$ and $w$ in the formula with the values you identified in the first step.
5. Calculate the Result: Perform the arithmetic operations (addition and multiplication) to find the final value of the perimeter.
6. Include the Units: The final answer must include the correct units of length. Since perimeter is a measure of distance, the units will be linear (like cm, m, inches, feet), not squared.

What If the Length and Width Have Different Units?

A common trick in math problems is to provide the dimensions of a shape in different units. For example, the length might be given in meters and the width in centimeters. You cannot substitute these different units directly into the formula. Doing so would be like adding apples and oranges; the result would be meaningless. You must first convert one of the measurements so that both are in the same unit.

You can choose which unit to convert to, but the problem might specify the desired unit for the final answer. Here is a table of common length conversions:

How Can You Find a Missing Side Length if You Know the Perimeter?

Sometimes, a problem will give you the total perimeter and the length of just one side, asking you to find the other. This requires using your algebra skills to rearrange the perimeter formula and solve for the unknown variable.

Let's start with our formula: $P=2l+2w$.

Suppose we know the perimeter ($P$) and the length ($l$), and we want to find the width ($w$). We need to isolate $w$ on one side of the equation.

1. Start with the formula: $P=2l+2w$
2. Subtract $2l$ from both sides: This step helps to get the term with $w$ by itself. $P-2l=2w$
3. Divide both sides by 2: This will leave $w$ all by itself. $\frac{P-2l}{2}=w$

So, the formula to find the width when you know the perimeter and length is:

Similarly, if you need to find the length ($l$) when you know the perimeter ($P$) and the width ($w$), the formula is:

How Does This Relate to the Perimeter of a Square?

A square is a special type of rectangle. It follows all the rules of a rectangle (four sides, four right angles, opposite sides equal), but it has one additional property: all four of its sides are equal in length.

Because a square is a rectangle, we can use the rectangle perimeter formula to find its perimeter. In a square, the length is equal to the width ($l=w$). Let's call the side length of the square $s$. So, $l=s$ and $w=s$.

If we substitute these into the rectangle formula $P=2l+2w$, we get:

Alternatively, using $P=2(l+w)$:

This gives us a simple, dedicated formula for the perimeter of a square.

This formula means the perimeter of a square is simply $4$ times the length of one of its sides. This makes sense, as you are just adding the same side length four times ($s+s+s+s$).

For example, if a square has a side length of $7$ inches, its perimeter is $P=4\times 7=28$ inches.

What Are Common Mistakes to Avoid?

When working with perimeter, a few common errors can trip students up. Being aware of these pitfalls is the best way to avoid them.

• Confusing Perimeter and Area: This is the most common mistake. Perimeter is the distance around a shape, while area is the space inside it. Remember that perimeter is measured in linear units (like inches, cm, miles) and area is measured in square units (like in², cm², miles²).
• Adding Only Two Sides: Some students mistakenly just add the length and width ($l+w$) and forget that a rectangle has four sides. This calculates only half the perimeter. Always remember to double this sum or use the $2l+2w$ formula.
• Forgetting to Convert Units: As shown in a previous section, if the length and width are in different units, you must convert them to the same unit before you begin any calculations. Your answer will be incorrect if you calculate with mixed units.
• Simple Calculation Errors: It's easy to make a small mistake in multiplication or addition, especially with larger numbers or decimals. Always take a moment to double-check your arithmetic before finalizing your answer. Writing out your steps clearly can help you catch these errors.
• Incorrectly Rearranging Formulas: When solving for a missing side, be careful with the order of operations. Remember to subtract before dividing when using the formula $w=(P-2l)/2$. A common mistake is to divide $P$ by $2$ first and then subtract $2l$, which is incorrect.

Quick Summary and Key Formulas

This lesson covered everything you need to know about the perimeter of a rectangle. Here are the most important points and formulas to remember:

Key Concepts

• Perimeter is the total distance around the outside of a two-dimensional shape.
• A rectangle has four sides, with opposite sides being equal in length.
• The sides are referred to as length ($l$) and width ($w$).
• Always ensure all measurements are in the same unit before calculating.
• Perimeter is a measure of length, so its units are linear (e.g., m, ft, cm), not squared.

Key Formulas

Perimeter of a Rectangle:

or

Perimeter of a Square (where $s$ is side length):

Finding a Missing Side (Width):