Solved Exercises on Finding Perimeter with Unknown Side Lengths in Grade 3

Master perimeter calculations with unknown side lengths: rectangles, squares, and other polygons through these 5 detailed exercises with step-by-step solutions.

Solution: Exercises 1 to 3
1 Finding unknown side in rectangle
Exercise 1
A rectangle has a perimeter of 24 cm and a width of 5 cm. What is the length of the rectangle?
Definition:

Rectangle perimeter formula: P = 2(length + width). To find unknown side: length = (P ÷ 2) - width.

Unknown side calculation method:
  1. Use the perimeter formula: P = 2(l + w)
  2. Substitute known values
  3. Solve for the unknown side
  4. Check your answer
Given
P = 24 cm, w = 5 cm
Formula
24 = 2(l + 5)
Result
l = 7 cm
Step 1: Write the formula

Perimeter = 2(length + width)

24 = 2(length + 5)

Step 2: Divide both sides by 2

24 ÷ 2 = 2(length + 5) ÷ 2

12 = length + 5

Step 3: Subtract 5 from both sides

12 - 5 = length + 5 - 5

7 = length

Step 4: Check your answer

Perimeter = 2(7 + 5) = 2(12) = 24 ✓

Final answer:

The length of the rectangle is 7 centimeters.

Applied rules:

• Rectangle perimeter: P = 2(l + w)

• Inverse operations: Use opposite operations to isolate unknown

• Verification: Always check your answer

2 Finding unknown side in square
Exercise 2
A square has a perimeter of 28 cm. What is the length of each side?
Definition:

Square perimeter formula: P = 4 × side length. To find side: side = P ÷ 4.

Step 1: Write the formula

Perimeter = 4 × side length

28 = 4 × side length

Step 2: Divide both sides by 4

28 ÷ 4 = (4 × side length) ÷ 4

7 = side length

Step 3: Check your answer

Perimeter = 4 × 7 = 28 ✓

Step 4: State the answer

Each side of the square is 7 cm long

Final answer:

Each side of the square is 7 centimeters long.

Applied rules:

• Square perimeter: P = 4s

• Division: To find one side, divide perimeter by 4

• Equal sides: All sides of a square are equal

3 Finding unknown side in triangle
Exercise 3
A triangle has a perimeter of 18 cm. Two of its sides measure 5 cm and 6 cm. What is the length of the third side?
Definition:

Triangle perimeter: P = side₁ + side₂ + side₃. To find unknown side: third side = P - (side₁ + side₂).

Step 1: Write the formula

Perimeter = side₁ + side₂ + side₃

18 = 5 + 6 + side₃

Step 2: Add known sides

5 + 6 = 11

Step 3: Subtract from perimeter

18 - 11 = 7

Step 4: Check your answer

5 + 6 + 7 = 18 ✓

Final answer:

The length of the third side is 7 centimeters.

Applied rules:

• Triangle perimeter: Sum of all three sides

• Subtraction: To find unknown side, subtract known sides from perimeter

• Verification: Add all sides to check

Key Concepts: Finding Unknown Side Lengths
\text{Rectangle: } l = \frac{P}{2} - w
Unknown Length
\text{Square: } s = \frac{P}{4}
Square Side
\text{Triangle: } \text{side}_3 = P - (\text{side}_1 + \text{side}_2)
Triangle Third Side
Rectangle
P = 2(l + w)
l = (P/2) - w
Square
P = 4s
s = P/4
Triangle
P = a + b + c
c = P - (a + b)
Regular Polygon
P = n × s
s = P/n
Inverse Operation
Solve Equations
Add ↔ Subtract, Multiply ↔ Divide
Verification
Check Answer
Substitute back to verify
Key definitions:

Perimeter: The total distance around the outside of a shape.

Unknown side: A side length that is not given in the problem but needs to be calculated.

Inverse operations: Operations that undo each other (addition and subtraction, multiplication and division).

Regular polygon: A polygon with all sides and angles equal.

Verification: Checking that your answer is correct by substituting it back into the original problem.

Complete methodology for unknown sides:
  1. Identify the shape: Determine what type of polygon you're working with
  2. Write the perimeter formula: Use the correct formula for that shape
  3. Substitute known values: Replace known variables with their values
  4. Solve for the unknown: Use inverse operations to isolate the unknown variable
  5. Check your answer: Substitute your answer back into the original formula
  6. State the final answer: Include the unit of measurement
Tip 1: Always remember that perimeter is the sum of ALL sides.
Tip 2: Use inverse operations to solve for unknown sides (add ↔ subtract, multiply ↔ divide).
Tip 3: Always check your answer by substituting it back into the perimeter formula.
Tip 4: Draw the shape and label known and unknown sides to visualize the problem.
Common patterns: Rectangle: P = 2(l + w), Square: P = 4s, Triangle: P = a + b + c.
Common mistakes: Forgetting to divide by 2 in rectangle problems, mixing up operations, not including units.
Essential rules for unknown sides:

• Perimeter = sum of all side lengths

• Rectangle: l = (P/2) - w or w = (P/2) - l

• Square: s = P/4

• Triangle: unknown side = P - (known side₁ + known side₂)

• Regular polygon: side = P ÷ number of sides

• Always verify your answer by substituting back

Solution: Exercises 4 to 5
4 Finding unknown side in irregular polygon
Exercise 4
A pentagon has a perimeter of 30 cm. Four of its sides measure 5 cm, 6 cm, 4 cm, and 7 cm. What is the length of the fifth side?
Definition:

Irregular polygon perimeter: P = sum of all sides. To find unknown side: unknown side = P - sum of known sides.

Step 1: Add all known sides

5 + 6 + 4 + 7 = 22 cm

Step 2: Subtract from total perimeter

30 - 22 = 8 cm

Step 3: Check your answer

5 + 6 + 4 + 7 + 8 = 30 ✓

Step 4: State the answer

The fifth side is 8 cm long

Final answer:

The length of the fifth side is 8 centimeters.

Applied rules:

• Irregular polygon: Perimeter = sum of all sides

• Subtraction: To find unknown side, subtract known sides from total

• Verification: Add all sides to confirm total perimeter

5 Word problem with unknown side
Exercise 5
A rectangular playground has a perimeter of 40 meters. If the width is 8 meters, what is the length? How much longer is the length than the width?
Definition:

Word problems: Real-life situations requiring mathematical calculations to solve.

Step 1: Use rectangle perimeter formula

P = 2(l + w)

40 = 2(l + 8)

Step 2: Divide both sides by 2

40 ÷ 2 = l + 8

20 = l + 8

Step 3: Subtract 8 from both sides

20 - 8 = l

12 = l

Step 4: Find the difference

Length - Width = 12 - 8 = 4 meters

Step 5: Check your answer

P = 2(12 + 8) = 2(20) = 40 ✓

Final answer:

The length of the playground is 12 meters. The length is 4 meters longer than the width.

Applied rules:

• Rectangle perimeter: P = 2(l + w)

• Inverse operations: Use opposite operations to solve

• Difference: Subtract to find how much longer

Complete Guide: Finding Unknown Side Lengths
\text{Unknown side} = \text{Perimeter} - \text{Known sides}
General Formula
\text{Rectangle: } l = \frac{P}{2} - w
Rectangle Formula
Key definitions:

Perimeter: The total distance around the boundary of a two-dimensional shape.

Unknown side: A missing measurement that can be calculated using the known perimeter and other side lengths.

Inverse operations: Mathematical operations that reverse the effect of another operation (addition reverses subtraction, multiplication reverses division).

Equation solving: The process of finding the value of an unknown variable by manipulating both sides of an equation.

Verification: The process of checking that a solution is correct by substituting it back into the original equation.

Complete problem-solving methodology:
  1. Read carefully: Identify what is given and what is unknown
  2. Choose formula: Select the appropriate perimeter formula for the shape
  3. Substitute values: Replace known values in the formula
  4. Isolate unknown: Use inverse operations to solve for the unknown
  5. Calculate: Perform the mathematical operations
  6. Verify: Check that your answer makes sense by substituting back
  7. State answer: Include units and answer the original question
Tip 1: Always start by writing down the perimeter formula for the specific shape.
Tip 2: When solving equations, whatever you do to one side, do to the other side.
Tip 3: Draw a picture and label what you know and what you need to find.
Tip 4: For rectangles, remember that opposite sides are equal in length.
Real-world applications: Fencing gardens, framing pictures, planning room layouts, building projects.
Algebra connection: This introduces basic algebra concepts of solving for unknowns in equations.
Essential rules for unknown sides:

• General principle: Unknown side = Perimeter - Sum of known sides

• Rectangle: l = (P ÷ 2) - w or w = (P ÷ 2) - l

• Square: s = P ÷ 4

• Triangle: third side = P - (side₁ + side₂)

• Regular polygon: side = P ÷ number of sides

• Verification: Always substitute back to confirm your answer

Questions & Answers

Question: I don't understand why we divide by 2 when solving for a rectangle's side. Can you explain?

Answer: When we use P = 2(l + w), we're adding both length and width, then multiplying by 2 because a rectangle has 2 lengths and 2 widths:

  • Perimeter = l + w + l + w = 2l + 2w = 2(l + w)
  • To solve for just one length or width, we first divide by 2 to get (l + w)
  • Then we subtract the known dimension to find the unknown

So if P = 24 and w = 5, then 24 = 2(l + 5), so 12 = l + 5, so l = 7.

Question: How do I know which operation to use when solving for unknown sides?

Answer: Use inverse operations to "undo" what's happening to the unknown:

  • If the unknown is being ADDED to something, SUBTRACT that amount
  • If the unknown is being MULTIPLIED by something, DIVIDE by that amount
  • If the unknown is being SUBTRACTED from something, ADD that amount
  • If the unknown is being DIVIDED by something, MULTIPLY by that amount

For example, if 24 = 2(l + 5), to get l alone, first divide by 2, then subtract 5.

Question: What if I have a shape with more than one unknown side? Can I still find them?

Answer: Generally, you need the same number of facts as unknowns:

  • With 1 unknown, you need 1 fact (like total perimeter)
  • With 2 unknowns, you usually need 2 facts (like total perimeter AND some relationship between sides)
  • For example, if you have a rectangle with 2 unknown sides but know they're equal (making it a square), then you can solve it

For now, focus on problems with just one unknown side. As you advance, you'll learn to handle multiple unknowns!