Complete Notes on Perimeter - Grade 5
📏 What is Perimeter?
Definition
Perimeter is the total distance around the outside of a closed shape.
Think of it as the length of a fence that goes all the way around a yard or the distance you would walk if you walked all the way around the edge of a shape.
🎯 Key Points to Remember:
🏠 Real-Life Examples:
- Fencing a yard: You need to know the perimeter to buy enough fencing
- Picture frames: The perimeter tells you how much frame material you need
- Running track: The perimeter is the distance around the track
- Garden borders: How much edging material you need for a garden bed
1. Perimeter with Whole Number Side Lengths
📐 Basic Concept
When all side lengths are whole numbers (like 5, 12, 20), finding perimeter is straightforward—just add all the sides together!
🔢 General Formula for ANY Shape:
\[ \text{Perimeter} = \text{Side}_1 + \text{Side}_2 + \text{Side}_3 + ... + \text{Side}_n \]
Add ALL the side lengths together!
📊 Common Shape Formulas:
| Shape | Formula | Description |
|---|---|---|
| Square | \(P = 4s\) | 4 times the side length |
| Rectangle | \(P = 2l + 2w\) OR \(P = 2(l + w)\) | 2 times length + 2 times width |
| Triangle | \(P = a + b + c\) | Add all three sides |
| Pentagon | \(P = s_1 + s_2 + s_3 + s_4 + s_5\) | Add all five sides |
| Hexagon | \(P = s_1 + s_2 + s_3 + s_4 + s_5 + s_6\) | Add all six sides |
| Regular Polygon | \(P = n \times s\) | Number of sides × side length |
✨ Step-by-Step Examples:
Example 1: Square
Problem: Find the perimeter of a square with side length = 8 cm
Solution:
\(P = 4s\)
\(P = 4 \times 8\)
\(P = 32\) cm
Answer: The perimeter is 32 cm
Example 2: Rectangle
Problem: Find the perimeter of a rectangle with length = 12 ft and width = 5 ft
Solution:
\(P = 2l + 2w\)
\(P = 2(12) + 2(5)\)
\(P = 24 + 10\)
\(P = 34\) ft
Answer: The perimeter is 34 ft
Example 3: Triangle
Problem: Find the perimeter of a triangle with sides = 7 m, 9 m, and 11 m
Solution:
\(P = a + b + c\)
\(P = 7 + 9 + 11\)
\(P = 27\) m
Answer: The perimeter is 27 m
Example 4: Irregular Polygon
Problem: Find the perimeter of a pentagon with sides = 4 in, 6 in, 5 in, 7 in, and 8 in
Solution:
\(P = 4 + 6 + 5 + 7 + 8\)
\(P = 30\) in
Answer: The perimeter is 30 inches
💡 Tips for Success:
- Always label your units (cm, m, ft, in, etc.)
- Make sure all sides are in the same unit before adding
- For shapes with equal sides (squares, regular polygons), use multiplication to save time
- Double-check that you've counted all sides
2. Perimeter with Decimal Side Lengths
📐 Basic Concept
When side lengths include decimals (like 5.5, 12.75, 3.2), the process is the SAME—just add all the sides together! You need to be careful with decimal addition.
🔢 Same Formulas Apply!
📝 Important Rules for Adding Decimals:
- Line up the decimal points when adding vertically
- Add zeros as placeholders if needed (5.2 = 5.20)
- The decimal point in the answer goes directly below the other decimal points
- Don't forget to include the decimal point in your final answer!
✨ Step-by-Step Examples:
Example 1: Square with Decimals
Problem: Find the perimeter of a square with side length = 6.5 cm
Solution:
\(P = 4s\)
\(P = 4 \times 6.5\)
\(P = 26.0\) cm or \(26\) cm
Answer: The perimeter is 26 cm
Example 2: Rectangle with Decimals
Problem: Find the perimeter of a rectangle with length = 15.8 m and width = 7.5 m
Solution:
\(P = 2l + 2w\)
\(P = 2(15.8) + 2(7.5)\)
\(P = 31.6 + 15.0\)
\(P = 46.6\) m
Answer: The perimeter is 46.6 m
Example 3: Triangle with Decimals
Problem: Find the perimeter of a triangle with sides = 8.4 ft, 10.25 ft, and 12.6 ft
Solution:
\(P = a + b + c\)
\(P = 8.4 + 10.25 + 12.6\)
Let's line up the decimals:
8.40
10.25
+12.60
______
31.25
\(P = 31.25\) ft
Answer: The perimeter is 31.25 ft
Example 4: Irregular Pentagon with Decimals
Problem: Find the perimeter of a pentagon with sides = 3.5 in, 4.2 in, 5.75 in, 6.1 in, and 4.8 in
Solution:
\(P = 3.5 + 4.2 + 5.75 + 6.1 + 4.8\)
Let's add step by step:
3.50
4.20
5.75
6.10
+4.80
_____
24.35
\(P = 24.35\) in
Answer: The perimeter is 24.35 inches
💡 Tips for Decimal Perimeters:
- Always line up decimal points when adding
- Add zeros as placeholders (6.5 becomes 6.50)
- Use a calculator to check your work
- Remember: \(2 \times 3.5 = 7.0\) or just \(7\)
- Don't forget to include the decimal point in your final answer!
3. Perimeter with Fractional Side Lengths
📐 Basic Concept
When side lengths are fractions or mixed numbers (like \(\frac{3}{4}\), \(2\frac{1}{2}\), \(5\frac{3}{8}\)), you still add all the sides—but you need to use fraction addition rules!
🔢 Same Formulas Apply!
📝 Important Rules for Adding Fractions:
🔄 Working with Mixed Numbers:
- Option 1: Convert mixed numbers to improper fractions, then add
- Option 2: Add whole numbers separately, then add fractions separately
✨ Step-by-Step Examples:
Example 1: Square with Fractions
Problem: Find the perimeter of a square with side length = \(\frac{3}{4}\) ft
Solution:
\(P = 4s\)
\(P = 4 \times \frac{3}{4}\)
\(P = \frac{12}{4}\)
\(P = 3\) ft
Answer: The perimeter is 3 ft
Example 2: Rectangle with Mixed Numbers
Problem: Find the perimeter of a rectangle with length = \(5\frac{1}{2}\) in and width = \(3\frac{1}{4}\) in
Solution:
\(P = 2l + 2w\)
\(P = 2(5\frac{1}{2}) + 2(3\frac{1}{4})\)
First, multiply:
\(2 \times 5\frac{1}{2} = 2 \times \frac{11}{2} = \frac{22}{2} = 11\) in
\(2 \times 3\frac{1}{4} = 2 \times \frac{13}{4} = \frac{26}{4} = 6\frac{2}{4} = 6\frac{1}{2}\) in
Then add:
\(P = 11 + 6\frac{1}{2} = 17\frac{1}{2}\) in
Answer: The perimeter is \(17\frac{1}{2}\) inches
Example 3: Triangle with Fractions
Problem: Find the perimeter of a triangle with sides = \(\frac{2}{3}\) m, \(\frac{3}{4}\) m, and \(\frac{5}{6}\) m
Solution:
\(P = a + b + c\)
\(P = \frac{2}{3} + \frac{3}{4} + \frac{5}{6}\)
Find common denominator (LCD = 12):
\(\frac{2}{3} = \frac{8}{12}\)
\(\frac{3}{4} = \frac{9}{12}\)
\(\frac{5}{6} = \frac{10}{12}\)
Add:
\(P = \frac{8}{12} + \frac{9}{12} + \frac{10}{12} = \frac{27}{12}\)
Simplify:
\(P = \frac{27}{12} = 2\frac{3}{12} = 2\frac{1}{4}\) m
Answer: The perimeter is \(2\frac{1}{4}\) m
Example 4: Rectangle with Mixed Numbers (Alternative Method)
Problem: Find the perimeter of a rectangle with length = \(4\frac{2}{5}\) cm and width = \(2\frac{3}{10}\) cm
Solution (Method 2 - Add whole and fraction parts separately):
\(P = 2l + 2w\)
\(P = 2(4\frac{2}{5}) + 2(2\frac{3}{10})\)
Multiply by 2:
\(2 \times 4\frac{2}{5} = 8\frac{4}{5}\)
\(2 \times 2\frac{3}{10} = 4\frac{6}{10} = 4\frac{3}{5}\)
Add whole numbers: \(8 + 4 = 12\)
Add fractions: \(\frac{4}{5} + \frac{3}{5} = \frac{7}{5} = 1\frac{2}{5}\)
Combine: \(12 + 1\frac{2}{5} = 13\frac{2}{5}\) cm
Answer: The perimeter is \(13\frac{2}{5}\) cm
💡 Tips for Fraction Perimeters:
- Always find the common denominator before adding
- Convert mixed numbers to improper fractions before multiplying
- Remember to simplify your final answer
- Check if your answer makes sense (it should be close to your estimate)
- You can add whole numbers and fractions separately if that's easier for you
📊 Quick Reference for Common Fractions:
| Mixed Number | Improper Fraction | Decimal |
|---|---|---|
| \(1\frac{1}{2}\) | \(\frac{3}{2}\) | 1.5 |
| \(2\frac{1}{4}\) | \(\frac{9}{4}\) | 2.25 |
| \(3\frac{1}{3}\) | \(\frac{10}{3}\) | 3.33... |
| \(4\frac{3}{4}\) | \(\frac{19}{4}\) | 4.75 |
| \(5\frac{2}{5}\) | \(\frac{27}{5}\) | 5.4 |
4. Perimeter of Figures on Grids
📐 Basic Concept
When a shape is drawn on a grid, you can find the perimeter by counting the unit squares along the edges of the shape.
📊 What is a Grid?
A grid is made up of squares that are all the same size. Each square has a specific unit (like 1 cm, 1 m, 1 in, etc.). The grid helps us measure shapes easily by counting squares.
🔢 Steps to Find Perimeter on a Grid:
📝 Important Tips for Grid Perimeters:
- Mark your starting point so you don't count the same side twice
- Count carefully - each edge of a grid square = 1 unit
- Check the scale - is each square 1 cm, 1 m, or something else?
- For rectilinear shapes (shapes with only straight edges and right angles), count each horizontal and vertical segment
- Diagonal lines are trickier - you may need to use the Pythagorean theorem
✨ Step-by-Step Examples:
Example 1: Rectangle on a Grid
Problem: A rectangle is drawn on a 1-cm grid. It is 5 squares long and 3 squares wide. Find the perimeter.
Solution:
Count the squares along each side:
- Top side: 5 cm
- Right side: 3 cm
- Bottom side: 5 cm
- Left side: 3 cm
\(P = 5 + 3 + 5 + 3 = 16\) cm
OR use the formula: \(P = 2(5) + 2(3) = 10 + 6 = 16\) cm
Answer: The perimeter is 16 cm
Example 2: Square on a Grid
Problem: A square is drawn on a 1-meter grid. Each side is 4 squares long. Find the perimeter.
Solution:
Each side = 4 meters
\(P = 4s = 4 \times 4 = 16\) m
Answer: The perimeter is 16 m
Example 3: Rectilinear Shape on a Grid (L-shape)
Problem: An L-shaped figure is drawn on a 1-cm grid. Count around the shape:
Starting from top-left corner, going clockwise:
- Top edge: 3 cm
- Down: 2 cm
- Right: 2 cm
- Down: 3 cm
- Left: 5 cm (3 + 2)
- Up: 5 cm (2 + 3)
\(P = 3 + 2 + 2 + 3 + 5 + 5 = 20\) cm
Answer: The perimeter is 20 cm
Example 4: Irregular Shape on a Grid
Problem: An irregular shape is drawn on a 1-inch grid. The sides measure: 2, 3, 4, 1, 2, and 4 inches going around the shape.
Solution:
\(P = 2 + 3 + 4 + 1 + 2 + 4 = 16\) in
Answer: The perimeter is 16 inches
💡 Pro Tips for Grid Perimeters:
- Use a pencil to mark each side as you count it
- Count carefully - don't skip any sides
- For complex shapes, write down each side length before adding
- Double-check that you've counted all the way around the shape
- Remember: The perimeter goes around the OUTSIDE of the shape only
- If sides have the same length, you can group them together
🎯 Special Case: Rectilinear Shapes
Rectilinear shapes are shapes made up of only horizontal and vertical line segments (like rectangles combined together).
Helpful Trick: For rectilinear shapes, the total of all horizontal segments equals the total of all vertical segments!
Example: If the top edges add up to 8 cm, the bottom edges also add up to 8 cm. If the left edges add up to 6 cm, the right edges also add up to 6 cm.
📊 Comparison Table: All Perimeter Types
| Type | Example | Key Strategy | Watch Out For |
|---|---|---|---|
| Whole Numbers | Sides: 5, 7, 9 Perimeter = 21 | Simple addition | Including all sides, correct units |
| Decimals | Sides: 5.5, 7.2, 9.8 Perimeter = 22.5 | Line up decimal points | Decimal point placement |
| Fractions | Sides: \(\frac{1}{2}\), \(\frac{3}{4}\), \(1\frac{1}{4}\) Perimeter = \(2\frac{1}{2}\) | Common denominators | Simplifying final answer |
| On Grids | Count squares: 3, 4, 3, 4 Perimeter = 14 units | Counting grid squares | Missing sides, correct scale |
📚 Quick Reference Formula Sheet
General Perimeter Formula
\[ P = \text{Sum of all side lengths} \]
Square
\[ P = 4s \]
where \(s\) = side length
Rectangle
\[ P = 2l + 2w \]
OR
\[ P = 2(l + w) \]
where \(l\) = length, \(w\) = width
Triangle
\[ P = a + b + c \]
where \(a\), \(b\), \(c\) are the three side lengths
Regular Polygon
\[ P = n \times s \]
where \(n\) = number of sides, \(s\) = length of one side
🎯 Key Reminders:
✏️ Practice Problem Types
Type 1: Find the Missing Side
If you know the perimeter and all sides except one, you can find the missing side!
Strategy: Perimeter - (sum of known sides) = missing side
Type 2: Compare Perimeters
Which shape has a greater perimeter?
Strategy: Calculate both perimeters and compare
Type 3: Real-World Problems
Example: How much fencing is needed for a rectangular garden?
Strategy: Find the perimeter using the dimensions given
Type 4: Shape on Grid
Count squares around the boundary of a shape.
Strategy: Mark your starting point and count carefully around the entire shape