Solved Exercises on Perimeter of Irregular Shapes in Grade 6

Master perimeter calculations: irregular shapes, step-by-step solutions, rules, methods, and practice problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 L-Shaped Figure
Exercise 1
Find the perimeter of an L-shaped figure with sides measuring:
Top horizontal: 8 cm
Right vertical: 5 cm
Bottom horizontal: 12 cm
Left vertical: 3 cm
Middle horizontal: 4 cm
Middle vertical: 2 cm
Definition:

Perimeter: The total distance around the outside of a shape

Irregular Shape: A shape with sides of different lengths and angles

Perimeter calculation method:
  1. Add up ALL side lengths around the shape
  2. Follow the path around the shape without missing any sides
  3. Count each side exactly once
All sides
8+5+12+3+4+2
Sum
34 cm
Step 1: Identify all sides

Trace the outside edge: 8 + 5 + 12 + 3 + 4 + 2

Step 2: Add all measurements

8 + 5 + 12 + 3 + 4 + 2 = 34

Step 3: State the unit

Total perimeter = 34 cm

Perimeter = 34 cm
Final answer:

The perimeter of the L-shaped figure is 34 cm

Applied rules:

• Perimeter Rule: Sum of all outer sides

• Measurement: Include units in final answer

• Accuracy: Count each side exactly once

8 cm 5 cm 12 cm 3 cm 4 cm 2 cm
2 T-Shaped Figure
Exercise 2
Find the perimeter of a T-shaped figure with sides measuring:
Top bar: 10 cm (horizontal), 2 cm (vertical)
Stem: 4 cm (horizontal), 8 cm (vertical)
Bottom: 10 cm (horizontal)
Definition:

T-Shape: A compound shape made of a rectangle on top of another rectangle

All sides
10+2+4+8+4+8+10+2
Sum
48 cm
Step 1: Trace the perimeter

Start from top left, go clockwise: 10 + 2 + 4 + 8 + 4 + 8 + 10 + 2

Step 2: Add all measurements

10 + 2 + 4 + 8 + 4 + 8 + 10 + 2 = 48

Step 3: Verify no sides missed

All 8 sides counted exactly once

Perimeter = 48 cm
Final answer:

The perimeter of the T-shaped figure is 48 cm

Applied rules:

• Systematic counting: Go around shape in order

• No repetition: Each side counted once

• Unit consistency: All measurements in same unit

10 cm 2 cm 8 cm 4 cm 10 cm
3 U-Shaped Figure
Exercise 3
Find the perimeter of a U-shaped figure with sides measuring:
Outer dimensions: 12 cm × 8 cm
Inner cutout: 6 cm × 4 cm
Calculate the perimeter of the remaining shape
Definition:

U-Shape: A rectangular frame with an inner rectangular cutout

Outer sides
12+8+12+8
Inner sides
6+4+6+4
Total
40+20=60 cm
Step 1: Calculate outer perimeter

12 + 8 + 12 + 8 = 40 cm

Step 2: Calculate inner perimeter

6 + 4 + 6 + 4 = 20 cm

Step 3: Add both perimeters

40 + 20 = 60 cm

Step 4: State the answer

Total perimeter = 60 cm

Perimeter = 60 cm
Final answer:

The perimeter of the U-shaped figure is 60 cm

Applied rules:

• Compound shapes: Include all outer and inner boundaries

• Complete tracing: Follow all edges including cutouts

• Accurate measurement: Count every side segment

12 cm 8 cm 6 cm 4 cm
Key Concepts: Rules, Laws, and Methods
Perimeter = Sum of all sides
Basic Formula
Rule 1
P = s₁ + s₂ + s₃ + ... + sₙ
Sum of all side lengths
Rule 2
P = 2(l + w)
Rectangle perimeter
Rule 3
P = 4s
Square perimeter
Key definitions:

Perimeter: Distance around the outside of a shape

Irregular Shape: Shape with unequal sides and angles

Regular Shape: Shape with equal sides and angles

Complete methodology:
  1. Identify the shape: Determine if regular or irregular
  2. List all sides: Write down each side length
  3. Add systematically: Sum all side lengths
  4. Check completeness: Ensure no sides missed
  5. State the unit: Include appropriate measurement unit
Tip 1: Start at one corner and trace around the shape in order.
Tip 2: Mark each side as you count it to avoid double-counting.
Tip 3: For complex shapes, break into simpler parts and add perimeters.
Tip 4: Always include units in your final answer.
Common errors: Missing sides, double-counting, forgetting units.
Exam preparation: Practice with various irregular shapes, focus on systematic counting.
Essential formulas to remember:

• Perimeter of any shape: P = sum of all sides

• Rectangle: P = 2(length + width)

• Square: P = 4 × side

• Triangle: P = side₁ + side₂ + side₃

Visual Guide: Perimeter Patterns
Understanding Perimeter Relationships
Visual comparison of different irregular shapes and their perimeter calculations:
Shape A: Simple L-shape
Shape B: Complex T-shape
Shape C: Frame-like U-shape

Analysis: The chart shows how different configurations affect perimeter values despite similar areas.

  • More complex shapes tend to have longer perimeters
  • Shapes with indentations have longer perimeters
  • Simple shapes have shorter perimeters for same area
Solution: Exercises 4 to 5
4 Cross-Shaped Figure
Exercise 4
Find the perimeter of a cross-shaped figure formed by a 6 cm × 2 cm rectangle crossing a 2 cm × 8 cm rectangle at the center. Both rectangles have the same width of 2 cm where they intersect.
Definition:

Cross-Shape: Two rectangles intersecting perpendicularly at their centers

Horizontal bar
6+2+6+2
Vertical bar
8+2+8+2
Overlap adjustment
Subtract overlapping parts
Step 1: Calculate horizontal rectangle perimeter

6 + 2 + 6 + 2 = 16 cm

Step 2: Calculate vertical rectangle perimeter

8 + 2 + 8 + 2 = 20 cm

Step 3: Account for intersection

Where they cross, subtract internal segments: 2 + 2 + 2 + 2 = 8 cm overlap

Step 4: Calculate total perimeter

16 + 20 - 8 = 28 cm

Perimeter = 28 cm
Final answer:

The perimeter of the cross-shaped figure is 28 cm

Applied rules:

• Intersection rule: Subtract overlapping internal segments

• Systematic counting: Track each unique boundary

• Geometric reasoning: Understand how shapes interact

6 cm 8 cm 2 cm
5 Staircase Figure
Exercise 5
Find the perimeter of a 4-step staircase shape where each step is 3 cm wide and 2 cm tall. The base is 12 cm wide and the total height is 8 cm.
Definition:

Staircase Shape: A series of rectangular steps forming a stepped pattern

Top horizontal
12 cm
Right vertical
8 cm
Bottom horizontal
12 cm
Step 1: Identify the outer boundary

The perimeter follows the outside edge: top, right, bottom, and left sides

Step 2: Calculate horizontal segments

Top: 12 cm, Bottom: 12 cm

Step 3: Calculate vertical segments

Right: 8 cm, Left: 8 cm (steps don't change outer vertical dimension)

Step 4: Sum all segments

12 + 8 + 12 + 8 = 40 cm

Step 5: Verify the result

For any rectangular outline, P = 2(length + width) = 2(12 + 8) = 40 cm

Perimeter = 40 cm
Final answer:

The perimeter of the staircase figure is 40 cm

Applied rules:

• Outer boundary rule: Only count the outside edges

• Step compensation: Internal steps don't change outer perimeter

• Verification: Check against rectangular formula

12 cm 8 cm 3×2 3×2
Comprehensive Guide: Laws, Methods, Rules, Definitions
P = s₁ + s₂ + s₃ + ... + sₙ
Perimeter Formula
Key definitions:

Perimeter: Total distance around the boundary of a 2D shape

Irregular Shape: Polygon with sides of different lengths and/or angles

Regular Shape: Polygon with equal sides and equal angles

Side Length: Measurement of one edge of a polygon

Bounding Path: The continuous line that forms the outer edge of a shape

Complete methodology:
  1. Shape Analysis: Examine the shape to identify all sides
  2. Boundary Tracing: Follow the outer edge systematically
  3. Side Identification: List each unique side length
  4. Measurement Verification: Ensure all sides are measured
  5. Summation: Add all side lengths together
  6. Unit Application: Express result with proper units
Tip 1: Use a ruler or straightedge to trace the perimeter and mark each side as you count it.
Tip 2: For complex shapes, draw arrows along each side to track your progress.
Tip 3: Break complex shapes into simpler parts and calculate each part separately.
Tip 4: Double-check by counting vertices - the number of sides equals the number of vertices in a polygon.
Tip 5: Remember that internal segments (like in cutouts) also contribute to the total perimeter.
Common errors: Missing internal boundaries, counting shared sides twice, forgetting units, skipping small segments.
Exam preparation: Practice with various irregular shapes, master systematic counting, memorize basic formulas.
Essential formulas and rules:

• General perimeter: P = sum of all side lengths

• Rectangle: P = 2(length + width)

• Square: P = 4 × side length

• Triangle: P = side₁ + side₂ + side₃

• Regular polygon: P = n × side length (where n = number of sides)

• Compound shapes: P = sum of all outer boundaries including internal cutouts

Special cases: For shapes with cutouts, include both outer and inner boundaries. For intersecting shapes, account for shared segments appropriately.
Problem-solving strategies: Draw auxiliary lines, break into regular shapes, use symmetry when possible, verify with alternative methods.

Questions & Answers

Question: I don't understand why in the cross-shaped figure we had to subtract the overlapping parts. Shouldn't we add everything?

Answer: Great question! The key is understanding what perimeter means:

  • Perimeter is the distance around the outside of the shape
  • When two rectangles intersect in a cross, the overlapping area creates internal boundaries
  • These internal boundaries are not part of the outside perimeter

Think of it like walking around the shape - you would walk along the outer edges only, not inside the crossbar. The overlapping parts represent segments that would be walked twice if we just added all sides, but the actual perimeter only includes each boundary segment once.

Example: If you have two rectangles intersecting, you calculate the full perimeter of each, then subtract the parts that become internal due to intersection.

Question: My child is struggling with complex irregular shapes. What's the best way to help them visualize the perimeter?

Answer: Here are effective visualization strategies:

  • Trace method: Have your child use their finger or a pencil to trace around the outside edge slowly
  • Color coding: Use different colored pencils to mark each side as it's counted
  • Arrow tracking: Draw arrows along each side in the order they're counted
  • Break it down: Separate complex shapes into simpler rectangles or triangles

Encourage your child to say each measurement out loud as they count it. This helps reinforce the process and reduces skipping sides. Also, practice with real-world examples like finding the perimeter of rooms, gardens, or playgrounds to make it more tangible.

The key is developing a systematic approach that works consistently across different shapes.

Question: How can I help students remember to include units in their perimeter answers?

Answer: Here are effective strategies to emphasize units:

  • Unit reminders: Always write "units" next to perimeter formulas
  • Context connections: Relate to real measurements (cm, m, ft)
  • Consistent modeling: Always include units in your examples
  • Mental check: Teach "Does my answer make sense?" with units

Create a simple checklist: "Find the lengths → Add them → Don't forget units!" Post this near students' work areas. You can also use mnemonics like "P-U-N" (Perimeter Units Needed) to help them remember.

Make it a habit to ask: "What unit should our answer be in?" before starting calculations.