How to Find the Missing Side of a Rectangle Without Perimeter: Step-by-Step Guide

If you know one side and the diagonal of a rectangle, you can use the Pythagorean theorem to find the missing side. The formula is:

Given the diagonal d and one side a, the other side b can be found using:

$$

b
=



d
2

-

a
2

Example

If the width a is 5 units and the diagonal d is 13 units:

$$

b
=


13

2
2

-
5

2
2

$$

b
=


169
-
25


=

144

=
12

So, the missing side b is 12 units.

If the area A and one side a are known, you can find the other side b using the formula:

$$

b
=


A


a

Example

If the area A is 60 square units and the width a is 5 units:

$$

b
=


60


5


=
12

So, the missing side b is 12 units.

Using these methods, you can find the missing side of a rectangle using either the diagonal or the area along with one known side.

For more detailed examples and calculations, you can visit resources like Calculator Soup, Khan Academy, and Math Stack Exchange.

If the area A and one side a are known, you can find the other side b using the formula:

$$

b
=


A


a

Example

If the area A is 60 square units and the width a is 5 units:

$$

b
=


60


5


=
12

So, the missing side b is 12 units.

Using these methods, you can find the missing side of a rectangle using either the diagonal or the area along with one known side.

For more detailed examples and calculations, you can visit resources like Calculator Soup, Khan Academy, and Math Stack Exchange.

Using these methods, you can find the missing side of a rectangle using either the diagonal or the area along with one known side.

For more detailed examples and calculations, you can visit resources like Calculator Soup, Khan Academy, and Math Stack Exchange.

Finding the missing side of a rectangle without using its perimeter involves understanding key mathematical principles and techniques. While the perimeter formula can be helpful, there are other effective ways to determine side lengths. This guide will explore:

• Using the Pythagorean theorem when the diagonal is known.
• Applying coordinate geometry for geometric insights.
• Employing algebraic methods for equations and unknowns.

We will start by reviewing the fundamental properties of rectangles and gradually move to practical example calculations, equipping you with the tools needed to solve for the missing side in various contexts.

Rectangles are fundamental geometric shapes characterized by their four right angles and opposite sides of equal length. To understand and solve for the missing side, it's crucial to know the following properties:

• Opposite Sides: In a rectangle, opposite sides are equal. If the length is \( l \) and the width is \( w \), then the opposite sides are also \( l \) and \( w \).
• Right Angles: All interior angles are 90 degrees, making it easy to apply the Pythagorean theorem.
• Diagonal Relationships: The diagonals of a rectangle are equal in length. Using the Pythagorean theorem, the length of the diagonal \( d \) can be calculated as:
  \( d = \sqrt{l^2 + w^2} \)
• Area: The area of a rectangle is calculated by multiplying its length and width:
  \( \text{Area} = l \times w \)

These properties provide the foundation for various methods to determine missing dimensions when certain measurements are known. We will use these properties in subsequent sections to solve for unknown sides without relying on the perimeter.

The Pythagorean theorem is a powerful tool for finding the missing side of a rectangle when the diagonal and one side are known. It states that in a right triangle, the square of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the other two sides.

To use the Pythagorean theorem to find a missing side of a rectangle:

1. Identify Known Values: Determine the lengths of the diagonal (\( d \)) and one side (either length \( l \) or width \( w \)).
2. Apply the Formula: The Pythagorean theorem can be expressed as:
   \( d^2 = l^2 + w^2 \)
3. Solve for the Missing Side: Rearrange the formula to solve for the unknown side:
   • If the length (\( l \)) is unknown:
     \( l = \sqrt{d^2 - w^2} \)
   • If the width (\( w \)) is unknown:
     \( w = \sqrt{d^2 - l^2} \)
4. Example: Suppose the diagonal is 13 units and the width is 5 units. To find the length:
   \( l = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \, \text{units} \)

This method provides an accurate way to determine the missing dimension using basic geometric principles, making it invaluable for various applications.

When the diagonal and one side of a rectangle are known, finding the missing side can be achieved using a straightforward application of the Pythagorean theorem. Here’s a step-by-step method to solve for the missing side:

1. Identify Known Values:
   • Diagonal (\( d \)): The length of the diagonal.
   • One Side (\( s \)): The known side, which could be either the length (\( l \)) or the width (\( w \)).
2. Set Up the Equation:

   Use the Pythagorean theorem \( d^2 = l^2 + w^2 \) to relate the diagonal and sides. Rearrange the equation based on whether the known side is the length or width.

   • Finding Length (\( l \)):
     \( l = \sqrt{d^2 - w^2} \)
   • Finding Width (\( w \)):
     \( w = \sqrt{d^2 - l^2} \)
3. Example Calculation:

   Suppose the diagonal (\( d \)) is 15 units and the width (\( w \)) is 9 units. To find the length (\( l \)):

   \( l = \sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144} = 12 \, \text{units} \)
4. Verification:

   To confirm the calculation, check that the computed side and the known side satisfy the original diagonal equation:

   \( d^2 = l^2 + w^2 \rightarrow 15^2 = 12^2 + 9^2 \rightarrow 225 = 144 + 81 \rightarrow 225 = 225 \)

   This confirms the solution is correct.

By following these steps, you can accurately determine the missing side of a rectangle when the diagonal and one side are provided.

To better understand how to find the missing side of a rectangle without using the perimeter, let's explore several example calculations using different known values.

Example 1: Known Diagonal and Width

1. Given:
   • Diagonal (\( d \)) = 17 units
   • Width (\( w \)) = 8 units
2. Find: Length (\( l \))
3. Calculation:
   \( l = \sqrt{d^2 - w^2} \)
   \( l = \sqrt{17^2 - 8^2} \)
   \( l = \sqrt{289 - 64} \)
   \( l = \sqrt{225} \)
   \( l = 15 \, \text{units} \)

Example 2: Known Diagonal and Length

1. Given:
   • Diagonal (\( d \)) = 20 units
   • Length (\( l \)) = 12 units
2. Find: Width (\( w \))
3. Calculation:
   \( w = \sqrt{d^2 - l^2} \)
   \( w = \sqrt{20^2 - 12^2} \)
   \( w = \sqrt{400 - 144} \)
   \( w = \sqrt{256} \)
   \( w = 16 \, \text{units} \)

Example 3: Known Length and Width

1. Given:
   • Length (\( l \)) = 9 units
   • Width (\( w \)) = 12 units
2. Find: Diagonal (\( d \))
3. Calculation:
   \( d = \sqrt{l^2 + w^2} \)
   \( d = \sqrt{9^2 + 12^2} \)
   \( d = \sqrt{81 + 144} \)
   \( d = \sqrt{225} \)
   \( d = 15 \, \text{units} \)

Example 4: Known Area and Width

1. Given:
   • Area = 120 square units
   • Width (\( w \)) = 10 units
2. Find: Length (\( l \))
3. Calculation:
   \( l = \frac{\text{Area}}{w} \)
   \( l = \frac{120}{10} \)
   \( l = 12 \, \text{units} \)

These examples illustrate how various known measurements can be used to find missing dimensions of a rectangle using straightforward mathematical methods. Each calculation builds on the basic principles outlined in previous sections.

Coordinate geometry provides a robust framework for finding the missing side of a rectangle by leveraging the coordinates of its vertices. This method involves plotting the rectangle on a coordinate plane and using distance formulas to determine side lengths. Follow these steps:

1. Plot the Rectangle:

   Place the rectangle on a coordinate plane. Assume you know the coordinates of three vertices, and need to find the coordinates of the fourth vertex. Label the vertices as \( A(x_1, y_1) \), \( B(x_2, y_2) \), \( C(x_3, y_3) \), and \( D(x_4, y_4) \), where the unknown vertex is \( D \).

2. Identify Known Coordinates:

   Determine the known coordinates of the vertices. For example, suppose you have:
   

   
   
   • \( A(0, 0) \)
   
   
   • \( B(a, 0) \)
   
   
   • \( C(a, b) \)
   
   
   • \( D(0, b) \) (to be determined)
   
   
   
   


3. Apply the Distance Formula:

   The distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) helps find the lengths of the sides:
   

   
   
   • Length:
     
     
     \( l = \sqrt{(a - 0)^2 + (0 - 0)^2} = a \)
     
     
     
   
   
   • Width:
     
     
     \( w = \sqrt{(a - a)^2 + (b - 0)^2} = b \)
     
     
     
   
   
   
   


4. Find the Missing Side:

   If any side’s coordinates are missing, use the known vertices and properties of rectangles (e.g., opposite sides are equal, right angles) to solve for the unknown coordinates:
   

   
   
   • To find \( D \), since \( D \) should align vertically with \( A \) and horizontally with \( C \):
     
     
     \( D = (0, b) \)
     
     
     
   
   
   
   


5. Example:

   If you know vertices \( A(0,0) \), \( B(6,0) \), and \( C(6,4) \), find \( D \):
   

   
   
   • Since \( A \) and \( D \) are vertically aligned and \( D \) should share the \( y \)-coordinate with \( C \):
     
     
     \( D = (0, 4) \)
     
     
     
   
   
   
   

Coordinate geometry allows for accurate determination of missing sides by translating geometric properties into coordinate-based calculations. This method is particularly useful in applications involving spatial positioning and graphical representations.

Algebraic methods offer a versatile approach to finding the missing side of a rectangle by using equations and relationships between sides, area, and diagonals. Here’s a step-by-step guide to solve for missing side lengths using algebra:

1. Identify Given Information:

   Start by noting down the known values. This can include:
   

   
   
   • Length (\( l \))
   
   
   • Width (\( w \))
   
   
   • Diagonal (\( d \))
   
   
   • Area (\( A \))
   
   
   
   


2. Set Up Relevant Equations:

   Depending on the given values, use the appropriate algebraic relationships:
   

   
   
   • Area Formula:
     
     
     \( A = l \times w \)
     
     
     
   
   
   • Diagonal Formula:
     
     
     \( d = \sqrt{l^2 + w^2} \)
     
     
     
   
   
   
   


3. Solve for the Missing Side:

   Use algebra to solve for the unknown side:
   

   
   
   • If Area and One Side are Known:
     
     
     \( w = \frac{A}{l} \) or \( l = \frac{A}{w} \)
     
     
     
   
   
   • If Diagonal and One Side are Known:
     
     
     \( l = \sqrt{d^2 - w^2} \) or \( w = \sqrt{d^2 - l^2} \)
     
     
     
   
   
   
   


4. Example Calculations:
   

   Example 1: Known Area and Width

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   Given:	Area (\( A \)) = 48 square units, Width (\( w \)) = 6 units
   Find:	Length (\( l \))
   Calculation:	\( l = \frac{A}{w} = \frac{48}{6} = 8 \, \text{units} \)

   
   

   Example 2: Known Diagonal and Length

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   Given:	Diagonal (\( d \)) = 10 units, Length (\( l \)) = 8 units
   Find:	Width (\( w \))
   Calculation:	\( w = \sqrt{d^2 - l^2} = \sqrt{10^2 - 8^2} = \sqrt{100 - 64} = \sqrt{36} = 6 \, \text{units} \)

   
   


5. Verification:

   Confirm your solution by plugging the found values back into the original equations to ensure they satisfy the known relationships.

Using algebraic methods simplifies the process of finding unknown side lengths by leveraging mathematical formulas and relationships, providing a clear and effective approach to solving these problems.

Finding the missing side of a rectangle without using the perimeter is a valuable skill in various real-world scenarios. This section explores practical applications and provides step-by-step methods for solving these problems.

Scenario 1: Construction and Landscaping

In construction and landscaping, precise measurements are crucial. If you have the diagonal and one side of a rectangular area, you can find the missing dimension to ensure accurate placement of materials or structures.

1. Problem: A rectangular garden has a diagonal of 50 meters and one side is 30 meters. Find the other side length.
2. Solution: Use the Pythagorean theorem.
   • Given: Diagonal (\( d \)) = 50 meters, Side (\( l \)) = 30 meters
   • Find: Width (\( w \))
   • Calculation:
     \( w = \sqrt{d^2 - l^2} \)
     \( w = \sqrt{50^2 - 30^2} \)
     \( w = \sqrt{2500 - 900} \)
     \( w = \sqrt{1600} \)
     \( w = 40 \, \text{meters} \)
3. Conclusion: The missing side length is 40 meters, ensuring correct dimensions for the garden layout.

Scenario 2: Home Improvement

When modifying room layouts or designing floor plans, knowing the missing side can help in space planning and furniture placement.

1. Problem: A room’s floor plan shows a diagonal of 25 feet and a width of 15 feet. Calculate the length.
2. Solution:
   • Given: Diagonal (\( d \)) = 25 feet, Width (\( w \)) = 15 feet
   • Find: Length (\( l \))
   • Calculation:
     \( l = \sqrt{d^2 - w^2} \)
     \( l = \sqrt{25^2 - 15^2} \)
     \( l = \sqrt{625 - 225} \)
     \( l = \sqrt{400} \)
     \( l = 20 \, \text{feet} \)
3. Conclusion: The room’s length is 20 feet, aiding in accurate interior design and space utilization.

Scenario 3: Material Estimation

Estimating material quantities for projects like tiling or fencing often requires knowing the dimensions of rectangular areas.

1. Problem: You need to tile a rectangular floor with a known area of 200 square feet and a width of 10 feet. Determine the length of the floor.
2. Solution: Use the area formula.
   • Given: Area (\( A \)) = 200 square feet, Width (\( w \)) = 10 feet
   • Find: Length (\( l \))
   • Calculation:
     \( l = \frac{A}{w} \)
     \( l = \frac{200}{10} \)
     \( l = 20 \, \text{feet} \)
3. Conclusion: The length is 20 feet, allowing accurate estimation of tiles required for the project.

These examples demonstrate the practical application of mathematical methods to solve for missing side lengths in various real-world contexts, facilitating better planning and decision-making.