Perimeter and Area - Sixth Grade
Complete Notes & Formulas
1. Perimeter
Definition
Perimeter is the DISTANCE AROUND a shape
The total length of all sides
Measured in linear units (cm, m, inches, feet)
Perimeter Formulas
| Shape | Formula | Alternative |
|---|---|---|
| Square | P = 4s | P = s + s + s + s |
| Rectangle | P = 2(l + w) | P = 2l + 2w |
| Triangle | P = a + b + c | Sum of all three sides |
| Any Polygon | P = Sum of all sides | Add all sides together |
Example: Find Perimeter of Rectangle
Problem: A rectangle has length 8 cm and width 5 cm. Find the perimeter.
P = 2(l + w)
P = 2(8 + 5)
P = 2(13)
P = 26 cm
Answer: 26 cm
2. Area - Basic Concept
Definition
Area is the amount of SPACE INSIDE a shape
The surface covered by the shape
Measured in SQUARE units (cm², m², in²)
Important: Perimeter = length units (cm, m), Area = SQUARE units (cm², m²)
3. Area of Rectangles and Squares
Rectangle
A = l × w
A = Area
l = length
w = width
Square
A = s²
or A = s × s
s = side length
Example
Problem: Find the area of a rectangle with length 12 m and width 7 m.
A = l × w
A = 12 × 7
A = 84 m²
Answer: 84 square meters
4. Area of Parallelograms
Formula
A = b × h
b = base (length of bottom side)
h = height (perpendicular distance between parallel sides)
Key Point: Height must be PERPENDICULAR (90°) to the base! Don't use the slant side.
Understanding
A parallelogram can be transformed into a rectangle by cutting and rearranging. The area formula is the same as a rectangle!
Example
Problem: A parallelogram has base 10 cm and height 6 cm. Find the area.
A = b × h
A = 10 × 6
A = 60 cm²
Answer: 60 cm²
5. Area of Triangles
Formula
A = ½ × b × h
or
A = (b × h) ÷ 2
b = base
h = height (perpendicular to base)
Understanding
A triangle is HALF of a parallelogram or rectangle. That's why we multiply by ½!
Height must be perpendicular (90°) to the base.
Example
Problem: A triangle has base 14 cm and height 8 cm. Find the area.
A = ½ × b × h
A = ½ × 14 × 8
A = ½ × 112
A = 56 cm²
Answer: 56 cm²
6. Area of Trapezoids
Formula
A = ½ × (b₁ + b₂) × h
or
A = [(b₁ + b₂) ÷ 2] × h
b₁ = first base (top parallel side)
b₂ = second base (bottom parallel side)
h = height (perpendicular distance between bases)
Understanding
Find the AVERAGE of the two bases, then multiply by height.
Think: (base₁ + base₂) ÷ 2 gives average base length.
Example
Problem: A trapezoid has bases 12 m and 8 m, and height 5 m. Find the area.
A = ½ × (b₁ + b₂) × h
A = ½ × (12 + 8) × 5
A = ½ × 20 × 5
A = ½ × 100
A = 50 m²
Answer: 50 m²
7. Area of Rhombuses
Formula (Using Diagonals)
A = ½ × d₁ × d₂
or
A = (d₁ × d₂) ÷ 2
d₁ = first diagonal
d₂ = second diagonal
Alternative Formula (Using Base and Height)
A = b × h
(same as parallelogram)
Example
Problem: A rhombus has diagonals 10 cm and 16 cm. Find the area.
A = ½ × d₁ × d₂
A = ½ × 10 × 16
A = ½ × 160
A = 80 cm²
Answer: 80 cm²
8. Area of Compound Figures
What is a Compound Figure?
A compound figure is made up of
TWO OR MORE simple shapes
combined together
Steps to Find Area
Step 1: Break the compound figure into simple shapes
Step 2: Find the area of each simple shape
Step 3: ADD all the areas together
Example
Problem: Find area of an L-shaped figure made of:
• Rectangle 1: 8 cm × 3 cm
• Rectangle 2: 5 cm × 4 cm
Step 1: Area of Rectangle 1 = 8 × 3 = 24 cm²
Step 2: Area of Rectangle 2 = 5 × 4 = 20 cm²
Step 3: Total Area = 24 + 20 = 44 cm²
Answer: 44 cm²
9. Area Between Two Shapes
Method
To find the area BETWEEN two shapes:
Area Between = Larger Area − Smaller Area
Example: Area Between Two Rectangles
Problem: A large rectangle is 10 m × 8 m. Inside it is a smaller rectangle 6 m × 4 m. Find the area between them.
Step 1: Area of large rectangle = 10 × 8 = 80 m²
Step 2: Area of small rectangle = 6 × 4 = 24 m²
Step 3: Area between = 80 − 24 = 56 m²
Answer: 56 m²
Quick Reference: All Area Formulas
| Shape | Area Formula | Variables |
|---|---|---|
| Square | A = s² | s = side |
| Rectangle | A = l × w | l = length, w = width |
| Parallelogram | A = b × h | b = base, h = height |
| Triangle | A = ½ × b × h | b = base, h = height |
| Trapezoid | A = ½(b₁ + b₂) × h | b₁, b₂ = bases, h = height |
| Rhombus | A = ½ × d₁ × d₂ | d₁, d₂ = diagonals |
10. Relationship Between Perimeter and Area
Important Concepts
• Same perimeter ≠ Same area
• Same area ≠ Same perimeter
• Perimeter and Area are INDEPENDENT
Example: Different Areas, Same Perimeter
Rectangle A: 6 × 2
Perimeter = 2(6 + 2) = 16 units
Area = 6 × 2 = 12 square units
Rectangle B: 5 × 3
Perimeter = 2(5 + 3) = 16 units
Area = 5 × 3 = 15 square units
Same perimeter (16), but DIFFERENT areas (12 vs 15)!
💡 Important Tips to Remember
✓ Perimeter = distance AROUND (add all sides)
✓ Area = space INSIDE (always in square units)
✓ Height must be PERPENDICULAR (90°) to base
✓ Triangle area = ½ of parallelogram
✓ Trapezoid: Average the two bases
✓ Rhombus uses diagonals (not base × height)
✓ Compound figures: Break into simple shapes
✓ Area between = Larger − Smaller
✓ Always include units! (cm, m for perimeter; cm², m² for area)
✓ Check your work - does the answer make sense?
🧠 Memory Tricks & Strategies
Perimeter vs Area:
"Perimeter is a FENCE around - Area is CARPET on the ground!"
Rectangle Area:
"Length times Width - that's how you get it right!"
Triangle Area:
"Triangle is half - so multiply base times height, then cut in half!"
Trapezoid Area:
"Add the bases, then divide by two, times the height - that's what you do!"
Compound Figures:
"Break it up, add it up - compound figures are not tough!"
Units:
"Perimeter walks around with feet, Area covers with square feet!"
Master Perimeter and Area! 📐 📏 □
Remember: Perimeter = around, Area = inside!