Solved Exercises on Finding Perimeter of Rectangle with Diagonal in Grade 3

Master finding perimeter of rectangles when given the diagonal: Pythagorean theorem, rectangle properties, perimeter calculation through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic rectangle
Exercise 1
A rectangle has length 3 units and width 4 units. What is its perimeter?
Definition:

Rectangle: A four-sided shape with opposite sides equal and all angles 90°

Perimeter: The total distance around the rectangle

Perimeter method:
  1. Add up all four sides: length + width + length + width
  2. Or use formula: Perimeter = 2 × (length + width)
Given
Length = 3, Width = 4
Formula
P = 2(l+w)
Result
P = 14
Step 1: Identify the dimensions

Length = 3 units, Width = 4 units

Step 2: Apply the perimeter formula

Perimeter = 2 × (length + width)

Perimeter = 2 × (3 + 4)

Step 3: Calculate the result

Perimeter = 2 × 7 = 14 units

Perimeter = 14 units
Final answer:

The perimeter of the rectangle is 14 units

Applied rules:

• Rectangle Property: Opposite sides are equal

• Perimeter Formula: P = 2(l+w) where l is length and w is width

• Order of Operations: Parentheses first, then multiplication

2 Diagonal introduction
Exercise 2
A rectangle has length 6 units, width 8 units, and diagonal 10 units. What is its perimeter?
Definition:

Diagonal: A line connecting opposite corners of a rectangle

Pythagorean Theorem: In a right triangle, a² + b² = c²

Given
Length = 6, Width = 8
Formula
P = 2(l+w)
Result
P = 28
Step 1: Verify the diagonal using Pythagorean theorem

Length² + Width² = Diagonal²

6² + 8² = 36 + 64 = 100 = 10² ✓

Step 2: Apply the perimeter formula

Perimeter = 2 × (length + width)

Perimeter = 2 × (6 + 8)

Step 3: Calculate the result

Perimeter = 2 × 14 = 28 units

Perimeter = 28 units
Final answer:

The perimeter of the rectangle is 28 units

Applied rules:

• Pythagorean Theorem: a² + b² = c² for right triangles

• Rectangle Property: Length and width form legs of right triangle with diagonal as hypotenuse

• Perimeter Formula: P = 2(l+w)

3 Find missing side
Exercise 3
A rectangle has one side 5 units, diagonal 13 units, and we need to find the other side to calculate perimeter.
Definition:

Pythagorean Theorem: a² + b² = c² where a,b are legs and c is hypotenuse

Missing Side: When one side and diagonal are known

Given
One side = 5, Diagonal = 13
Pythagorean
5² + b² = 13²
Result
b = 12
Step 1: Set up the Pythagorean equation

We know: One side = 5, Diagonal = 13

Using: a² + b² = c²

So: 5² + b² = 13²

Step 2: Solve for the unknown side

25 + b² = 169

b² = 169 - 25 = 144

b = √144 = 12

Step 3: Calculate the perimeter

Now we know both sides: 5 and 12

Perimeter = 2 × (5 + 12) = 2 × 17 = 34

Step 4: Verify with Pythagorean theorem

5² + 12² = 25 + 144 = 169 = 13² ✓

Perimeter = 34 units
Final answer:

The perimeter of the rectangle is 34 units

Applied rules:

• Pythagorean Theorem: a² + b² = c²

• Algebraic Manipulation: Solving for unknown variable

• Square Root: Finding the positive root of a perfect square

Rules and methods, laws,...
\(P = 2(l + w)\)
Perimeter of Rectangle
Pythagorean Theorem
\(a^2 + b^2 = c^2\)
For right triangles
Rectangle Diagonal
\(l^2 + w^2 = d^2\)
Length, width, diagonal
Area of Rectangle
\(A = l \times w\)
Length times width
Key definitions:

Rectangle: Four-sided shape with opposite sides equal and all angles 90°

Perimeter: Total distance around the rectangle

Diagonal: Line connecting opposite corners

Pythagorean Theorem: Relationship in right triangles

Complete methodology:
  1. Identify given information: Which sides/diagonal are known
  2. Draw the rectangle: Label known values and mark what you need to find
  3. Apply Pythagorean theorem: If needed to find missing side
  4. Calculate perimeter: Use P = 2(l+w)
  5. Verify your answer: Check if it makes sense
Tip 1: Remember that length and width form the legs of a right triangle with the diagonal as the hypotenuse.
Tip 2: The diagonal always divides the rectangle into two identical right triangles.
Tip 3: When solving for a missing side, always take the positive square root since lengths are positive.
Tip 4: Always verify your answer by checking that the Pythagorean theorem holds.
Common errors: Forgetting to multiply by 2 in perimeter formula, taking negative square root, misidentifying which values are the legs vs hypotenuse.
Memory aids: "Perimeter is the fence around the yard" and "a-squared plus b-squared equals c-squared" (Pythagorean theorem).
Formulas to know by heart:

• Perimeter of rectangle: \(P = 2(l + w)\)

• Area of rectangle: \(A = l \times w\)

• Pythagorean theorem: \(a^2 + b^2 = c^2\)

• Rectangle diagonal: \(l^2 + w^2 = d^2\)

Solution: Exercises 4 to 5
4 Complex diagonal problem
Exercise 4
A rectangle has a diagonal of 15 units. If one side is 9 units, find the other side and the perimeter.
Definition:

Right Triangle Legs: The two shorter sides of a right triangle

Hypotenuse: The longest side of a right triangle (opposite the right angle)

Given
Diagonal = 15, One side = 9
Pythagorean
9² + b² = 15²
Result
b = 12, P = 42
Step 1: Set up the Pythagorean relationship

In a rectangle, the diagonal creates a right triangle with the two sides

Known: Diagonal = 15 units (hypotenuse), one side = 9 units (leg)

Unknown: Other side = ? units (other leg)

Using: a² + b² = c²

So: 9² + b² = 15²

Step 2: Solve for the unknown side

81 + b² = 225

b² = 225 - 81 = 144

b = √144 = 12 units

Step 3: Calculate the perimeter

Now we have both sides: 9 units and 12 units

Perimeter = 2 × (length + width) = 2 × (9 + 12) = 2 × 21 = 42 units

Step 4: Verify the solution

Check: 9² + 12² = 81 + 144 = 225 = 15² ✓

Other side = 12 units, Perimeter = 42 units
Final answer:

The other side is 12 units, and the perimeter is 42 units

Applied rules:

• Pythagorean Theorem: a² + b² = c² for right triangles

• Rectangle Property: The diagonal forms a right triangle with the two sides

• Algebraic Manipulation: Solving for unknown variables

5 Word problem application
Exercise 5
A rectangular garden has a diagonal path of 20 meters. If the length of the garden is 16 meters, what is the width and the perimeter of the garden?
Definition:

Word Problem Strategy: Identify key information and translate to mathematical terms

Real-world Application: Connecting math concepts to practical scenarios

Given
Diagonal = 20m, Length = 16m
Pythagorean
16² + w² = 20²
Result
w = 12m, P = 56m
Step 1: Identify the given information

Rectangular garden with diagonal path = 20 meters

Length of garden = 16 meters

Need to find: Width and Perimeter

Step 2: Apply the Pythagorean theorem

The diagonal of the rectangle creates a right triangle with length and width

So: Length² + Width² = Diagonal²

16² + Width² = 20²

Step 3: Solve for the width

256 + Width² = 400

Width² = 400 - 256 = 144

Width = √144 = 12 meters

Step 4: Calculate the perimeter

Perimeter = 2 × (length + width) = 2 × (16 + 12) = 2 × 28 = 56 meters

Step 5: Verify the solution

Check: 16² + 12² = 256 + 144 = 400 = 20² ✓

Width = 12 meters, Perimeter = 56 meters
Final answer:

The width of the garden is 12 meters, and the perimeter is 56 meters

Applied rules:

• Word Problem Analysis: Extract relevant information from context

• Pythagorean Theorem: a² + b² = c² for right triangles

• Rectangle Properties: Opposite sides are equal, all angles are 90°

Complete Guide: Laws, Methods, Rules, Definitions, Expressions
\(P = 2(l + w)\)
Perimeter of Rectangle
Key definitions:

Rectangle: A quadrilateral with four right angles (90°) and opposite sides equal in length

Perimeter: The total distance around the boundary of a shape

Diagonal: A straight line connecting two opposite corners of a rectangle

Pythagorean Theorem: In a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides

Complete methodology:
  1. Understand the problem: Identify what's given and what needs to be found
  2. Draw a diagram: Sketch the rectangle and label known values
  3. Identify relationships: Recognize that the diagonal creates a right triangle with length and width
  4. Apply Pythagorean theorem: Use a² + b² = c² to find missing dimensions
  5. Calculate perimeter: Use P = 2(l + w)
  6. Verify solution: Check calculations and ensure the answer makes sense
Tip 1: Always draw a picture when solving geometry problems. It helps visualize the relationships between different parts of the rectangle.
Tip 2: In rectangle problems with diagonals, remember that the diagonal, length, and width form a right triangle.
Tip 3: When solving for a missing side using the Pythagorean theorem, always take the positive square root since lengths are positive.
Tip 4: Always verify your answer by substituting back into the Pythagorean theorem.
Rectangle Properties: Opposite sides are equal, all angles are 90°, diagonals are equal in length and bisect each other.
Pythagorean Triples: Common sets of numbers that satisfy the Pythagorean theorem (e.g., 3-4-5, 5-12-13, 8-15-17).
Common Errors: Confusing the diagonal with a side length, forgetting to multiply by 2 in the perimeter formula, taking the negative square root.
Essential formulas:

• Perimeter of rectangle: P = 2(l + w)

• Area of rectangle: A = l × w

• Pythagorean theorem: a² + b² = c²

• Rectangle diagonal relationship: l² + w² = d²

• Distance between opposite corners: d = √(l² + w²)

Rectangle Perimeter with Diagonal Infographic
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Rectangle Structure

• 4 sides: 2 lengths (l) + 2 widths (w)

• 4 right angles (90°)

• Diagonal creates 2 right triangles

• Diagonal length: d = √(l² + w²)

Perimeter Calculation

• Perimeter = l + w + l + w

• Perimeter = 2l + 2w

• Perimeter = 2(l + w)

• Units: same as side measurements

Pythagorean Theorem

• Applies to right triangles

• a² + b² = c²

• For rectangle: l² + w² = d²

• c is always the hypotenuse

Problem-Solving Workflow
1
Identify Given Values
2
Draw Diagram
3
Apply Formula
4
Calculate
5
Verify Answer
Visual Representation

Rectangle with length (l), width (w), and diagonal (d)

Right triangle formed: legs = l & w, hypotenuse = d

Perimeter = 2(l + w)

Remember: Diagonal splits rectangle into 2 congruent right triangles!

Questions & Answers

Question: Why do we need to use the Pythagorean theorem to find the perimeter of a rectangle when we have the diagonal? Can't we just add up the sides directly?

Answer: Great question! You can only add up the sides directly if you already know both the length and width. But when you're given the diagonal and only one side, you don't know the other side yet. That's where the Pythagorean theorem comes in!

  • If you know both length and width: Perimeter = 2(length + width) - no need for Pythagorean theorem
  • If you know diagonal and one side: Use Pythagorean theorem to find the missing side, then calculate perimeter

Think of it this way: the diagonal, length, and width form a right triangle inside the rectangle. The Pythagorean theorem (a² + b² = c²) helps us find the missing measurement.

Example: If diagonal is 5 and one side is 3, we use 3² + b² = 5² to find the other side (which is 4), then calculate perimeter as 2(3+4) = 14.

Question: I don't understand why the diagonal creates a right triangle. Can you explain that?

Answer: Yes! Let me explain why the diagonal creates a right triangle in a rectangle:

  • A rectangle has four corners, and each corner is a 90-degree angle (right angle)
  • When you draw a diagonal line from one corner to the opposite corner, it connects to two sides that meet at a right angle
  • This creates a triangle where one angle is the corner of the rectangle (90 degrees), making it a right triangle

Think of it like this: If you have a rectangle piece of paper and fold it along the diagonal, the two sides that were perpendicular (meeting at a 90-degree angle) still form that same right angle with the diagonal crease.

The diagonal becomes the hypotenuse of this right triangle, while the length and width of the rectangle become the two legs of the triangle.

Question: When I solve for the missing side using the Pythagorean theorem, sometimes I get a decimal. Should I round it?

Answer: Whether to round depends on the problem's instructions and context:

  • If the problem asks for an exact answer, keep it as a radical (e.g., √10) or fraction
  • If the problem specifies rounding, follow those instructions (nearest tenth, hundredth, etc.)
  • If the problem involves real-world measurements, rounding may be appropriate

However, for many rectangle problems in 3rd grade, you'll work with "nice" numbers that result in perfect squares. For example: 3-4-5, 5-12-13, 8-15-17 are common Pythagorean triples where all sides are whole numbers.

If you do get a decimal, try checking your work first - make sure you didn't make a calculation error. Sometimes rechecking the arithmetic reveals a perfect square that was missed.