Area - Fifth Grade
Complete Notes & Formulas
What is Area?
Area is the amount of space inside a two-dimensional shape. It measures how much surface a shape covers.
Area = Space Inside a Shape
Measured in Square Units
Key Points About Area
1. Area is TWO-DIMENSIONAL
Measured in square units: cm², m², ft², in², etc.
2. Different from Perimeter
Perimeter = distance around (linear)
Area = space inside (square)
3. Always use SQUARE UNITS
5 cm × 3 cm = 15 cm² (NOT 15 cm)
4. Think of it as counting squares
How many unit squares fit inside?
1. Area of Squares and Rectangles
A. Area of a Rectangle
Area = Length × Width
A = l × w
or A = l × b (breadth)
Example 1: Rectangle
Problem: Find the area of a rectangle with length 8 cm and width 5 cm.
Given:
Length = 8 cm
Width = 5 cm
Solution:
A = l × w
A = 8 × 5
A = 40 cm²
Answer: 40 square centimeters
B. Area of a Square
Area = Side × Side
A = s²
(Because all sides are equal)
Example 2: Square
Problem: Find the area of a square with side 6 m.
Given: Side = 6 m
Solution:
A = s²
A = 6²
A = 6 × 6 = 36 m²
Answer: 36 square meters
Remember: Area is ALWAYS in square units (cm², m², ft², in²)!
2. Area of Rectangles with Fractions
How to Multiply Fractions
To multiply fractions:
Multiply numerators × Multiply denominators
a/b × c/d = (a × c)/(b × d)
Example 1: Both Dimensions are Fractions
Problem: Find the area of a rectangle with length 3/4 m and width 2/5 m.
Solution:
A = l × w
A = 3/4 × 2/5
A = (3 × 2)/(4 × 5)
A = 6/20
A = 3/10 m² (simplified)
Answer: 3/10 square meters
Example 2: One Whole Number, One Fraction
Problem: Find the area of a rectangle with length 6 cm and width 2/3 cm.
Solution:
A = l × w
A = 6 × 2/3
A = (6 × 2)/3
A = 12/3 = 4 cm²
Answer: 4 square centimeters
3. Area of Rectangles with Fractions and Mixed Numbers
Steps to Find Area with Mixed Numbers
Step 1: Convert mixed numbers to improper fractions
Step 2: Multiply the fractions
Step 3: Simplify if needed
Step 4: Convert back to mixed number if needed
Converting Mixed Numbers
Mixed Number → Improper Fraction
a b/c = (a × c + b)/c
Example: 2 1/3 = (2 × 3 + 1)/3 = 7/3
Example 1: Both Mixed Numbers
Problem: Find the area of a rectangle with length 3 1/2 ft and width 2 1/4 ft.
Step 1: Convert to improper fractions
3 1/2 = (3 × 2 + 1)/2 = 7/2
2 1/4 = (2 × 4 + 1)/4 = 9/4
Step 2: Multiply
A = 7/2 × 9/4
A = (7 × 9)/(2 × 4)
A = 63/8
Step 3: Convert to mixed number
63 ÷ 8 = 7 remainder 7
A = 7 7/8 ft²
Answer: 7 7/8 square feet
Example 2: Alternate Method (Distributive Property)
Problem: Find the area: 2 1/2 × 1 1/3
Break apart and multiply:
2 1/2 = (2 + 1/2)
1 1/3 = (1 + 1/3)
Multiply each part:
2 × 1 = 2
2 × 1/3 = 2/3
1/2 × 1 = 1/2
1/2 × 1/3 = 1/6
Add all parts:
2 + 2/3 + 1/2 + 1/6 = 3 1/3
Answer: 3 1/3 square units
4. Area of Compound Figures
What is a Compound Figure?
A compound figure (also called composite figure) is a shape made up of two or more simple shapes put together.
Steps to Find Area of Compound Figures
Step 1: Break the compound figure into simple shapes (rectangles, squares, triangles)
Step 2: Find the dimensions of each simple shape
Step 3: Calculate the area of each simple shape
Step 4: Add all the areas together
Example: L-Shaped Figure
Problem: Find the area of this L-shaped figure.
Visual:
Rectangle 1: 10 cm × 4 cm (horizontal part)
Rectangle 2: 6 cm × 3 cm (vertical part)
Step 1: Break into 2 rectangles
Step 2: Find area of each
Rectangle 1: A = 10 × 4 = 40 cm²
Rectangle 2: A = 6 × 3 = 18 cm²
Step 3: Add them together
Total Area = 40 + 18 = 58 cm²
Answer: 58 square centimeters
Formula for Compound Figures:
Total Area = Area₁ + Area₂ + Area₃ + ...
5. Area Between Two Rectangles
What Does This Mean?
This is the area of a frame or border - the space between an outer rectangle and an inner rectangle.
Formula
Area Between = Area of Outer Rectangle − Area of Inner Rectangle
Example: Picture Frame
Problem: A picture frame has outer dimensions 12 in × 10 in. The picture inside is 8 in × 6 in. Find the area of the frame.
Step 1: Find area of outer rectangle
Area (outer) = 12 × 10 = 120 in²
Step 2: Find area of inner rectangle
Area (inner) = 8 × 6 = 48 in²
Step 3: Subtract
Area of frame = 120 − 48 = 72 in²
Answer: 72 square inches
Tip: Always subtract the INNER area from the OUTER area!
6. Area of Figures on Grids
How to Find Area on a Grid
Method 1: Count the Squares
Count how many unit squares fit inside the shape
Method 2: Use Formula
Count length and width, then multiply
Example 1: Rectangle on Grid
Problem: Find the area of a rectangle on a grid. Each square = 1 cm².
The rectangle is 5 squares long and 3 squares wide.
Method 1 (Counting):
Count all squares: 15 squares
Area = 15 cm²
Method 2 (Formula):
Length = 5 units, Width = 3 units
A = 5 × 3 = 15 cm²
Answer: 15 square centimeters
Example 2: Irregular Shape on Grid
Problem: Find the area of an irregular shape on a grid.
Strategy: Break into simple rectangles
Count squares in each rectangle
Add them together
OR simply count all the unit squares!
7-9. Area and Perimeter Word Problems
Key Differences
| Concept | Perimeter | Area |
|---|---|---|
| Measures | Distance around | Space inside |
| Units | Linear (m, cm, ft) | Square (m², cm², ft²) |
| Operation | Add sides | Multiply l × w |
| Example Use | Fence, border, frame | Carpet, paint, tile |
Example 1: Word Problem with Whole Numbers
Problem: A rectangular garden is 15 m long and 8 m wide. How much fencing is needed? How much area can be planted?
Fencing = Perimeter
P = 2(l + w) = 2(15 + 8) = 2(23) = 46 m
Planting area = Area
A = l × w = 15 × 8 = 120 m²
Answers: 46 m of fencing, 120 m² for planting
Example 2: With Fractions
Problem: A rectangular table is 4 1/2 ft long and 2 1/3 ft wide. Find the area and perimeter.
Area:
Convert: 4 1/2 = 9/2, 2 1/3 = 7/3
A = 9/2 × 7/3 = 63/6 = 10 1/2 ft²
Perimeter:
P = 2(4 1/2 + 2 1/3)
P = 2(6 5/6) = 13 2/3 ft
Answers: Area = 10 1/2 ft², Perimeter = 13 2/3 ft
Example 3: With Decimals
Problem: A rectangle is 6.5 cm long and 3.2 cm wide. Find area and perimeter.
Area:
A = 6.5 × 3.2 = 20.8 cm²
Perimeter:
P = 2(6.5 + 3.2) = 2(9.7) = 19.4 cm
Answers: Area = 20.8 cm², Perimeter = 19.4 cm
Quick Reference: Area Formulas
| Shape | Area Formula | Example |
|---|---|---|
| Rectangle | A = l × w | 8 × 5 = 40 |
| Square | A = s² | 6² = 36 |
| Compound Figure | Sum of parts | A₁ + A₂ |
| Area Between | Outer − Inner | 120 − 48 = 72 |
💡 Important Tips to Remember
✓ Area = Space INSIDE a shape
✓ Always use SQUARE UNITS (cm², m², ft², in²)
✓ Rectangle: Multiply length × width
✓ Square: Side × Side or s²
✓ With fractions: Convert mixed numbers to improper fractions first
✓ Compound figures: Break apart and add areas
✓ Area between: Outer − Inner
✓ On grids: Count squares or use length × width
✓ Area ≠ Perimeter! Different concepts!
✓ Fencing = Perimeter, Carpet/Paint = Area
🧠 Memory Tricks
Area vs Perimeter:
Area = Inside (like flooring or carpet)
Perimeter = Around (like a fence)
Rectangle Formula:
"Length times Width = How much space inside"
Square Units:
"Square the unit! cm × cm = cm²"
Mixed Numbers:
"Make improper before you multiply!"
Compound Figures:
"Break it, Find it, Add it!"
Area Between:
"Big minus Small = Frame on the wall!"
Master Area! ▭ ⬜ 📐
Area measures space inside - practice calculating it every day!