Perimeter and Area

How do we measure around a shape and the space inside it? Perimeter and area are fundamental concepts for measuring two-dimensional figures!

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What Is Perimeter?

Perimeter is the distance around the outside of a shape.

Think: Walking around the edge of a shape - how far would you walk?

Units: Linear units (inches, feet, meters, cm, etc.)

Formula idea: Add up all the side lengths!

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What Is Area?

Area is the amount of space inside a shape.

Think: How many square tiles would cover the shape?

Units: Square units (square inches, square feet, square meters, cm², etc.)

Formula idea: Length × Width for rectangles, variations for other shapes!

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Key Difference: Perimeter vs Area

Perimeter:

• Distance AROUND
• One-dimensional (length)
• Units: inches, cm, feet, meters
• Like a fence around a yard

Area:

• Space INSIDE
• Two-dimensional (length × width)
• Units: square inches, cm², square feet, m²
• Like carpet covering a floor

Different measurements for different purposes!

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Rectangle Perimeter

Perimeter of rectangle = 2 × length + 2 × width

Or: P = 2l + 2w

Or: P = 2(l + w)

Example: Rectangle with length 8 cm, width 5 cm

P = 2(8) + 2(5) P = 16 + 10 P = 26 cm

Or: P = 2(8 + 5) = 2(13) = 26 cm

All sides added: 8 + 5 + 8 + 5 = 26 cm

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Rectangle Area

Area of rectangle = length × width

A = l × w

Example: Rectangle with length 8 cm, width 5 cm

A = 8 × 5 A = 40 cm²

Read as "40 square centimeters"

Think: 8 rows of 5 square centimeters = 40 squares total

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Square Perimeter

A square has all four sides equal.

Perimeter of square = 4 × side

P = 4s

Example: Square with side 6 inches

P = 4 × 6 P = 24 inches

Simple: just multiply side length by 4!

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Square Area

Area of square = side × side

A = s²

Example: Square with side 6 inches

A = 6 × 6 A = 36 square inches A = 36 in²

This is why we call it "squared"!

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Triangle Perimeter

Perimeter of triangle = sum of all three sides

P = a + b + c

Example: Triangle with sides 5 cm, 7 cm, 8 cm

P = 5 + 7 + 8 P = 20 cm

Just add all three sides!

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Triangle Area

Area of triangle = 1/2 × base × height

A = (1/2)bh or A = bh/2

Important: Height must be PERPENDICULAR to the base!

Example: Triangle with base 10 cm, height 6 cm

A = (1/2) × 10 × 6 A = (1/2) × 60 A = 30 cm²

Think: A triangle is half a rectangle!

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Understanding Triangle Height

Height (altitude):

• Perpendicular distance from base to opposite vertex
• Forms a right angle with the base
• May be inside, outside, or on the side of the triangle

Any side can be the base!

• Just make sure height is perpendicular to that base

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Parallelogram Area

Area of parallelogram = base × height

A = bh

Example: Parallelogram with base 9 m, height 4 m

A = 9 × 4 A = 36 m²

Note: Height is perpendicular to base, not the slanted side!

Perimeter: Add all four sides (not just base × 2!)

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Trapezoid Area

Area of trapezoid = 1/2 × (base₁ + base₂) × height

A = (1/2)(b₁ + b₂)h

Trapezoid: Quadrilateral with exactly one pair of parallel sides

Example: Trapezoid with bases 8 ft and 12 ft, height 5 ft

A = (1/2)(8 + 12) × 5 A = (1/2)(20) × 5 A = 10 × 5 A = 50 ft²

Think: Average of the two bases, times the height!

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Circle Circumference

Circumference is the perimeter of a circle.

C = 2πr (using radius) C = πd (using diameter)

π (pi) ≈ 3.14 or π ≈ 22/7

Example: Circle with radius 7 cm

C = 2π(7) C = 14π C ≈ 14 × 3.14 C ≈ 43.96 cm

Or: C = πd = π(14) ≈ 43.96 cm

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Circle Area

Area of circle = πr²

A = πr²

Example: Circle with radius 5 inches

A = π(5)² A = π(25) A = 25π A ≈ 25 × 3.14 A ≈ 78.5 in²

Remember: Radius squared, then multiply by π!

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Finding Radius from Diameter

Diameter = 2 × radius Radius = diameter ÷ 2

Example: Circle with diameter 20 m

Radius = 20 ÷ 2 = 10 m

Then: C = πd = π(20) ≈ 62.8 m A = πr² = π(10)² = 100π ≈ 314 m²

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Composite Figures

Composite figure: Made of multiple shapes combined.

Strategy:

1. Break into simpler shapes
2. Find area of each part
3. Add (or subtract) areas

Example: L-shaped figure

Can be: Two rectangles added Or: Large rectangle minus small rectangle

Both methods give same answer!

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Composite Example: L-Shape

L-shape:

• Horizontal part: 10 × 3 = 30 units²
• Vertical part: 4 × 5 = 20 units²
• Total: 30 + 20 = 50 units²

Or:

• Large rectangle: 10 × 8 = 80 units²
• Missing corner: 6 × 5 = 30 units²
• Total: 80 - 30 = 50 units²

Same answer both ways!

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Composite Example: Rectangle with Semicircle

Figure: Rectangle 8 × 6 with semicircle on top (diameter 8)

Rectangle area: A = 8 × 6 = 48 units²

Semicircle area: Radius = 8 ÷ 2 = 4 Full circle: A = π(4)² = 16π Semicircle: A = 16π ÷ 2 = 8π ≈ 25.12 units²

Total area: 48 + 25.12 ≈ 73.12 units²

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Units Matter!

Perimeter: Linear units

• 20 feet, 30 cm, 15 m

Area: Square units

• 50 ft², 100 cm², 75 m²

Converting units:

• 1 foot = 12 inches
• 1 square foot = 144 square inches (12 × 12)

Always include units in your answer!

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Real-World Applications

Perimeter:

• Fencing a yard (how much fence?)
• Framing a picture
• Edging a garden bed
• Running track length

Area:

• Carpeting a room (how much carpet?)
• Painting a wall (how much paint?)
• Tiling a floor (how many tiles?)
• Seeding a lawn (how much seed?)

Different uses, different measurements!

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Finding Missing Dimensions

If you know perimeter:

Rectangle perimeter = 40 cm, length = 12 cm, find width

P = 2l + 2w 40 = 2(12) + 2w 40 = 24 + 2w 16 = 2w w = 8 cm

If you know area:

Rectangle area = 63 m², length = 9 m, find width

A = l × w 63 = 9 × w w = 7 m

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Scaling and Area

If you double the dimensions, area quadruples!

Original square: side = 2, area = 4 Doubled square: side = 4, area = 16

Why? Area = s²

• Original: 2² = 4
• Doubled: 4² = 16 (multiplied by 4!)

If you triple dimensions, area is multiplied by 9! (3²)

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Comparing Shapes with Same Perimeter

Different shapes can have same perimeter but different areas!

Example: Both have perimeter 24 units

Rectangle 8 × 4:

• Perimeter: 2(8 + 4) = 24
• Area: 8 × 4 = 32 units²

Rectangle 6 × 6 (square):

• Perimeter: 4(6) = 24
• Area: 6 × 6 = 36 units²

Square has larger area with same perimeter!

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Comparing Shapes with Same Area

Different shapes can have same area but different perimeters!

Example: Both have area 36 units²

Rectangle 9 × 4:

• Area: 9 × 4 = 36
• Perimeter: 2(9 + 4) = 26

Square 6 × 6:

• Area: 6 × 6 = 36
• Perimeter: 4(6) = 24

Square has smaller perimeter with same area!

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Irregular Shapes on Grid

For shapes on a grid:

Method 1: Count whole squares and partial squares

• Whole squares = 1 unit² each
• Two halves = 1 unit²

Method 2: Enclose in rectangle, subtract uncovered parts

Estimate partial squares when needed!

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Formulas Summary

Rectangle:

• Perimeter: P = 2l + 2w or P = 2(l + w)
• Area: A = lw

Square:

• Perimeter: P = 4s
• Area: A = s²

Triangle:

• Perimeter: P = a + b + c
• Area: A = (1/2)bh

Parallelogram:

• Area: A = bh

Trapezoid:

• Area: A = (1/2)(b₁ + b₂)h

Circle:

• Circumference: C = 2πr or C = πd
• Area: A = πr²

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Common Mistakes to Avoid

❌ Mistake 1: Confusing perimeter and area

• Perimeter = around, Area = inside

❌ Mistake 2: Wrong units

• Perimeter: feet, Area: square feet

❌ Mistake 3: Using slant height instead of perpendicular height

• Height must be perpendicular to base!

❌ Mistake 4: Forgetting to square the radius in circle area

• A = πr² not πr

❌ Mistake 5: Adding areas when you should subtract

• In composite figures, check if parts overlap

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Problem-Solving Strategy

To find perimeter:

1. Identify the shape
2. Find all side lengths
3. Add them up
4. Include linear units

To find area:

1. Identify the shape
2. Find needed measurements (base, height, radius, etc.)
3. Use correct formula
4. Include square units

For composite figures:

1. Break into simpler shapes
2. Find area of each
3. Add or subtract as needed
4. Check work makes sense

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Estimation Skills

Estimate before calculating:

Perimeter: About how many sides added? Area: About how many squares fit inside?

Rounding:

• π ≈ 3 for quick estimates
• Round dimensions to nearest whole number

Check: Does final answer match estimate?

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Quick Reference

Perimeter:

• Distance around
• Add all sides
• Linear units

Area:

• Space inside
• Length × width (for rectangles)
• Square units

Key formulas:

• Rectangle: P = 2(l + w), A = lw
• Square: P = 4s, A = s²
• Triangle: A = (1/2)bh
• Circle: C = 2πr, A = πr²

Remember:

• Height perpendicular to base
• Radius is half diameter
• Include units in answer!

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Practice Tips

Tip 1: Draw and label figures

• Visual helps understanding
• Mark what you know

Tip 2: Write formula first

• Then substitute values
• Reduces errors

Tip 3: Check units

• Convert if needed
• Square units for area!

Tip 4: Estimate first

• Helps catch big errors
• Builds number sense

Tip 5: Practice with real objects

• Measure room perimeter
• Calculate floor area
• Makes it concrete!

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Summary

Perimeter and area measure different aspects of shapes:

Perimeter:

• Distance around the outside
• Add all side lengths
• Uses linear units (cm, m, ft, in)
• Applications: fencing, framing, borders

Area:

• Amount of space inside
• Uses formulas based on shape
• Uses square units (cm², m², ft², in²)
• Applications: flooring, painting, covering

Key concepts:

• Different shapes, different formulas
• Height must be perpendicular
• Composite figures: break into parts
• Scaling changes area dramatically
• Always include proper units

Essential formulas:

• Rectangle/Square: A = lw or s²
• Triangle: A = (1/2)bh
• Circle: A = πr², C = 2πr
• Perimeter: sum of all sides

Master these and you can measure any two-dimensional space!