Rectangle perimeter: The distance around the rectangle. Formula: P = 2 × (length + width)
- Add the length and width
- Multiply the sum by 2
- State the unit (cm, m, etc.)
Length = 8 cm, Width = 5 cm
Perimeter = 2 × (length + width)
P = 2 × (8 + 5)
P = 2 × 13 = 26 cm
The perimeter of the rectangle is 26 cm.
• Perimeter formula: P = 2 × (length + width)
• Order of operations: Calculate parentheses first
• Units: Always include units in the final answer
Triangle perimeter: The distance around the triangle. Formula: P = side₁ + side₂ + side₃
Side 1 = 6 cm, Side 2 = 8 cm, Side 3 = 10 cm
Perimeter = side₁ + side₂ + side₃
P = 6 + 8 + 10
P = 24 cm
The perimeter of the triangle is 24 cm.
• Triangle perimeter formula: P = side₁ + side₂ + side₃
• Always add: All three sides of the triangle
• Units: Always include units in the final answer
Combined shape perimeter: Add the perimeters of individual parts, excluding shared sides.
Rectangle perimeter = 2 × (length + width)
P = 2 × (10 + 4) = 2 × 14 = 28 cm
Triangle perimeter = side₁ + side₂ + side₃
P = 4 + 5 + 5 = 14 cm
Don't count the shared side (width of rectangle where triangle base is attached)
Total perimeter = Rectangle P + Triangle P - 2 × shared side
P = 28 + 14 - 2×4 = 42 - 8 = 34 cm
The perimeter of the combined shape is 34 cm.
• Shared sides: Don't count sides shared between shapes
• Individual calculations: Calculate each shape separately first
• Addition principle: Add individual perimeters after removing shared parts
Rectangle perimeter: The distance around the rectangle. Formula: P = 2 × (length + width)
Length = 12 cm, Width = 7 cm
Perimeter = 2 × (length + width)
P = 2 × (12 + 7)
P = 2 × 19 = 38 cm
The perimeter of the rectangle is 38 cm.
• Perimeter formula: P = 2 × (length + width)
• Order of operations: Calculate parentheses first
• Units: Always include units in the final answer
Triangle perimeter: The distance around the triangle. Formula: P = side₁ + side₂ + side₃
Side 1 = 9 cm, Side 2 = 12 cm, Side 3 = 15 cm
Perimeter = side₁ + side₂ + side₃
P = 9 + 12 + 15
P = 36 cm
The perimeter of the triangle is 36 cm.
• Triangle perimeter formula: P = side₁ + side₂ + side₃
• Always add: All three sides of the triangle
• Units: Always include units in the final answer
Combined shape perimeter: Add the perimeters of individual parts, excluding shared sides.
Rectangle perimeter = 2 × (length + width)
P = 2 × (15 + 5) = 2 × 20 = 40 cm
Triangle perimeter = side₁ + side₂ + side₃
P = 5 + 6 + 6 = 17 cm
Don't count the shared side (width of rectangle where triangle base is attached)
Total perimeter = Rectangle P + Triangle P - 2 × shared side
P = 40 + 17 - 2×5 = 57 - 10 = 47 cm
The perimeter of the combined shape is 47 cm.
• Shared sides: Don't count sides shared between shapes
• Individual calculations: Calculate each shape separately first
• Addition principle: Add individual perimeters after removing shared parts
Rectangle perimeter: The distance around the rectangle. Formula: P = 2 × (length + width)
Length = 20 cm, Width = 10 cm
Perimeter = 2 × (length + width)
P = 2 × (20 + 10)
P = 2 × 30 = 60 cm
The perimeter of the rectangle is 60 cm.
• Perimeter formula: P = 2 × (length + width)
• Order of operations: Calculate parentheses first
• Units: Always include units in the final answer
Triangle perimeter: The distance around the triangle. Formula: P = side₁ + side₂ + side₃
Side 1 = 12 cm, Side 2 = 16 cm, Side 3 = 20 cm
Perimeter = side₁ + side₂ + side₃
P = 12 + 16 + 20
P = 48 cm
The perimeter of the triangle is 48 cm.
• Triangle perimeter formula: P = side₁ + side₂ + side₃
• Always add: All three sides of the triangle
• Units: Always include units in the final answer
Combined shape perimeter: Add the perimeters of individual parts, excluding shared sides.
Rectangle perimeter = 2 × (length + width)
P = 2 × (18 + 6) = 2 × 24 = 48 cm
Triangle perimeter = side₁ + side₂ + side₃
P = 6 + 7 + 7 = 20 cm
Don't count the shared side (width of rectangle where triangle base is attached)
Total perimeter = Rectangle P + Triangle P - 2 × shared side
P = 48 + 20 - 2×6 = 68 - 12 = 56 cm
The perimeter of the combined shape is 56 cm.
• Shared sides: Don't count sides shared between shapes
• Individual calculations: Calculate each shape separately first
• Addition principle: Add individual perimeters after removing shared parts
Advanced combined shape perimeter: Add the perimeters of individual parts, excluding shared sides. When two triangles are added to opposite sides of a rectangle, the straight sides cancel out.
Rectangle perimeter = 2 × (length + width)
P = 2 × (14 + 8) = 2 × 22 = 44 cm
Each triangle has sides = 8 cm, 10 cm, 10 cm
Perimeter of two triangles = 2 × (8 + 10 + 10) = 56 cm
Don't count the shared sides (both widths of rectangle where triangles are attached)
Total perimeter = Rectangle P + 2×Triangle P - 4×shared side
P = 44 + 56 - 4×8 = 100 - 32 = 68 cm
The perimeter of the advanced combined shape is 68 cm.
• Shared sides: Don't count sides shared between shapes
• Multiple attachments: Account for all shared sides
• Complex combinations: Carefully track which sides to exclude
Perimeter: The distance around the outside of a shape
Rectangle: A four-sided shape with four right angles
Triangle: A three-sided polygon with three angles
- Identify the shape(s): Determine if rectangle, triangle, or combination
- Select the formula: Rectangle: P = 2(l + w), Triangle: P = a + b + c
- Apply the formula: Substitute known values
- Calculate: Perform the arithmetic
- Check units: Include appropriate measurement units
• Rectangle perimeter: P = 2(length + width)
• Triangle perimeter: P = side₁ + side₂ + side₃
• Alternative rectangle: P = length + width + length + width
• Combined shapes: P = Sum of perimeters - 2×shared sides
Questions & Answers
Question: Why do we multiply by 2 in the rectangle perimeter formula?
Answer: Great question! A rectangle has two pairs of equal sides:
- Two sides with length = length
- Two sides with length = width
So the total perimeter is: length + width + length + width
We can group these: (length + length) + (width + width) = 2×length + 2×width
Using the distributive property: 2×length + 2×width = 2×(length + width)
That's why the formula is P = 2(l + w) - it's just a shorter way to add all four sides!
Question: How do I know which sides to subtract when combining shapes?
Answer: The key is to identify which sides are internal to the combined shape:
- Count only the outer edges of the combined shape
- Sides that are inside the shape (shared between the rectangle and triangle) should not be counted
- Visualize walking around the outside of the shape - only count the distances you walk along the outside
For example, when a triangle is attached to a rectangle, the base of the triangle and the side of the rectangle where it attaches become internal. So you calculate the rectangle perimeter, add the triangle perimeter, then subtract the lengths of the shared sides.
Drawing the shape and tracing the outer path can help visualize which sides to include.
Question: Do all triangles have the same perimeter formula?
Answer: Yes, all triangles use the same perimeter formula regardless of their type:
- Equilateral triangle: All sides equal (P = s + s + s = 3s)
- Isosceles triangle: Two sides equal (P = a + a + b = 2a + b)
- Scalene triangle: All sides different (P = a + b + c)
The general formula is always P = side₁ + side₂ + side₃, which works for all triangles.
The specific formulas like P = 3s for equilateral triangles are just simplified versions of the general formula when we know the sides are equal.
Visual Guide: Perimeter of Rectangle and Triangle
Complete Summary: Perimeter of Rectangle and Triangle
Perimeter: The total distance around the outside of a shape.
Rectangle: A four-sided shape with four right angles (90°), opposite sides are equal and parallel.
Triangle: A three-sided polygon with three angles and three sides.
Dimensions: Length (longer side) and width (shorter side) for rectangles; side lengths for triangles.
Rectangle Perimeter: P = 2(l + w) or P = l + w + l + w
Triangle Perimeter: P = a + b + c (add all three sides)
Combined Shapes: P = Sum of individual perimeters - 2 × shared sides
Triangle Types: Equilateral (3 equal sides), Isosceles (2 equal sides), Scalene (all different)
- Identify the Shape: Determine if rectangle, triangle, or combination
- Measure Dimensions: Find length, width, or side lengths
- Select Formula: Choose appropriate perimeter formula
- Substitute Values: Replace variables with actual measurements
- Calculate: Perform the mathematical operations
- Include Units: Always add measurement units (cm, m, in, ft, etc.)
Simple Rectangle: Length = 5 cm, Width = 3 cm
P = 2(5 + 3) = 2(8) = 16 cm
Simple Triangle: Sides = 4 cm, 5 cm, 6 cm
P = 4 + 5 + 6 = 15 cm
Combined Shape: Rectangle (8×4) + Triangle (sides 4, 5, 5)
P = 2(8+4) + (4+5+5) - 2×4 = 24 + 14 - 8 = 30 cm
- Memorization Trick: "2 times length plus width" for rectangles
- Three Sides: Remember to add all three sides for triangles
- Unit Consistency: All measurements must be in the same unit
- Common Mistake: Forgetting to multiply by 2 in rectangle formula
- Shared Sides: Don't count sides shared between shapes twice
- Verification: Add all sides individually to check your answer
- Rectangles: Opposite sides equal, P = 2(length + width)
- Triangles: Always add all three sides, P = a + b + c
- Perimeter: Always measures the outside boundary
- Combined Shapes: Subtract shared sides from total
- Practice: Work with various dimensions to build fluency
- Visualization: Draw shapes to understand shared sides