Perimeter & Area — Square, Rectangle, Triangle, and More
Perimeter & Area — Square, Rectangle, Triangle, and More. Practice questions to deepen understanding of perimeter and area of squares, rectangles, triangles, and other shapes. Online math practice with full solutions and step-by-step explanations.
Perimeter & Area — Square, Rectangle, Triangle, and More. Compute perimeter and area for common shapes. Apply formulas in word problems and visual exercises.
📐 Simple rectangle:
A rectangle has length 8 cm and width 5 cm.
What is the perimeter of the rectangle?
💡 Rectangle perimeter formula
📐 Formula:
Perimeter = 2×(length + width)
🔢 Calculation:
Length = 8 cm, width = 5 cm
Perimeter = 2×(8 + 5) = 2×13 = 26 cm
✨ Why?
A rectangle has 4 sides: 8 + 8 + 5 + 5 = 26 cm
🟦 Rectangle area:
A rectangle has length 6 cm and width 4 cm.
What is the area of the rectangle?
💡 Rectangle area formula
📐 Formula:
Area = length × width
🔢 Calculation:
Length = 6 cm, width = 4 cm
Area = 6 × 4 = 24 cm²
📏 Units: Area is always in cm²
🟨 Square:
A square has side length 7 cm.
What is the perimeter of the square?
💡 Square perimeter formula
📐 Formula:
Perimeter = 4 × side
🔢 Calculation:
Side = 7 cm
Perimeter = 4 × 7 = 28 cm
✨ Why?
A square has 4 equal sides: 7 + 7 + 7 + 7 = 28 cm
🟦 Square area:
A square has side length 5 cm.
What is the area of the square?
💡 Square area formula
📐 Formula:
Area = side × side = side²
🔢 Calculation:
Side = 5 cm
Area = 5 × 5 = 25 cm²
🔲 Comparing rectangles:
Rectangle A: length 10 cm, width 3 cm
Rectangle B: length 8 cm, width 4 cm
Which rectangle has a larger perimeter?
💡 Comparing perimeters
🔴 Rectangle A:
Perimeter = 2×(10 + 3) = 26 cm
🔵 Rectangle B:
Perimeter = 2×(8 + 4) = 24 cm
⚖️ 26 > 24 → Rectangle A is larger by 2 cm!
📦 Composite shape:
The shape consists of two identical squares.
Each square: side = 4 cm
What is the total area?
💡 Composite shape area
📐 One square: 4 × 4 = 16 cm²
📐 Two squares: 16 + 16 = 32 cm²
✨ Composite shape = sum of individual areas!
🏃 Running track:
Rectangular track: length 50 m, width 20 m.
A runner went around the track once.
How many meters did the runner go??
💡 Real-life problem
🏃 "Went around the track" = covered the full perimeter!
🔢 Perimeter = 2×(50 + 20) = 2×70 = 140 m
🎨 Picture frame:
Picture: 30 cm × 20 cm
We want to place a frame around it.
How many cm of frame do we need?
💡 Practical application
🖼️ A frame goes all the way around = perimeter!
🔢 Perimeter = 2×(30 + 20) = 100 cm
🏡 Square garden:
A square garden with side = 12 m.
We want to place a fence around it.
How many meters of fence do we need?
💡 Practical problem
🏡 A fence around the garden = perimeter of a square
🔢 Perimeter = 4 × 12 = 48 m
📱 Phone screen:
Screen: 15 cm × 8 cm
What is the area of the screen?
💡 Summary Part 1
📱 Screen = rectangle → Area = length × width
🔢 Area = 15 × 8 = 120 cm²
🎓 Formulas learned:
Perimeter of rectangle = 2×(length + width)
Area of rectangle = length × width
📐 Rectangle with variable:
A rectangle has length \(2x\) and width 5.
Write an expression for the perimeter. the rectangle.
💡 Perimeter with a variable
📐 Formula: Perimeter = 2×(length + width)
🔢 Substitution:
Length = 2x, width = 5
Perimeter = 2×(2x + 5) = 4x + 10
✨ Expanding brackets:
2×(2x + 5) = 2×2x + 2×5 = 4x + 10
🟦 Area with variable:
A rectangle has length \(3x\) and width 4.
Write an expression for the area. the rectangle.
💡 Area with a variable
📐 Formula: Area = length × width
🔢 Calculation:
Length = 3x, width = 4
Area = 3x × 4 = 12x
🟨 Square with variable:
A square has side length \(x\).
Write an expression for the perimeter. the square.
💡 Square perimeter with a variable
📐 Formula: Perimeter = 4 × side
🔢 Substitution:
Side = x
Perimeter = 4 × x = 4x
✨ Remember: no multiplication sign needed between a number and a variable!
🟦 Square area with variable:
A square has side length \(x\).
Write an expression for the area. the square.
💡 Square area with a variable
📐 Formula: Area = side × side = side²
🔢 Calculation:
Side = x
Area = x × x = x²
✨ x² is called "x squared" — it means x multiplied by itself.
📐 Special rectangle:
A rectangle has length \(x+3\) and width \(x\).
Write a simplified expression for the perimeter.
💡 Combining like terms
📐 Perimeter:
= 2×(length + width)
= 2×(x + 3 + x)
= 2×(2x + 3)
= 4x + 6
🔢 Or the long way:
= 2(x + 3) + 2x = 2x + 6 + 2x = 4x + 6
🔢 Substituting a value:
A rectangle has length \(2x\) and width 5.
If \(x=4\)what is the perimeter?
💡 Substituting a value
📐 Step 1 — Find the expression:
Perimeter = 4x + 10
🔢 Step 2 — Substitute x = 4:
= 4×4 + 10 = 16 + 10 = 26 cm
📊 Area with substitution:
A rectangle has length \(5x\) and width 3.
If \(x=2\) what is the area?
💡 Area with substitution
📐 Step 1 — Expression:
Area = 5x × 3 = 15x
🔢 Step 2 — Substitute x = 2:
= 15 × 2 = 30 cm²
🎯 Complex expression:
A rectangle has length \(3x+2\) and width \(x\).
Write a simplified expression for the perimeter..
💡 Simplifying a complex expression
📐 Perimeter:
= 2×(3x + 2 + x)
= 2×(4x + 2)
= 8x + 4
🔢 Or:
= 2(3x + 2) + 2x = 6x + 4 + 2x = 8x + 4
🔷 Two expressions:
Rectangle I: length \(2x\), width 5
Rectangle II: length \(x+4\), width \(0.5x\)
Write an expression for the perimeter. every rectangle.
💡 Two rectangles
🔴 Rectangle I:
Perimeter = 2(2x + 5) = 4x + 10 ✓
🔵 Rectangle II:
Perimeter = 2(x + 4 + 0.5x) = 2(1.5x + 4) = 3x + 8
Note: the two perimeters are different expressions!
🏆 Summary Part 2:
Rectangle: length \(x+6\), width \(2x\)
a. Write a simplified expression for the perimeter.
b. Calculate the perimeter when \(x=3\)
💡 Comprehensive summary
📐 Part a — expression:
Perimeter = 2(x + 6 + 2x) = 2(3x + 6) = 6x + 12
🔢 Part b — substitution:
x = 3 → 6×3 + 12 = 18 + 12 = 30 cm
🪜 Staircase:
A stair-shaped figure. Each step: 4 cm width, 4 cm height.
there is 5 stairs/steps. What is the perimeter of the shape?
'
💡 Correct perimeter calculation for stairs
🪜 Correct count:
📐 Horizontal lines:
• Bottom: one full line = 4×5 = 20 cm
• Top steps: 5 lines of 4 cm each = 20 cm
📐 Vertical lines:
• Side: one tall line = 4×5 = 20 cm
• Steps: 5 lines of 4 cm each = 20 cm
🔢 Total perimeter = 20 + 20 + 20 + 20 = 80 cm
📦 L-shape:
An L-shaped figure made of two rectangles:
Rectangle 1: 12 cm × 6 cm
rectangle 2: 6 cm × 6 cm
What is the total area?
💡 Area of an L-shape
📐 Dividing into parts:
Rectangle 1: 12 × 6 = 72 cm²
Rectangle 2: 6 × 6 = 36 cm²
🔢 Total area:
= 72 + 36 = 108 cm²
🔲 Square with a hole:
Large square: side 10 cm
Small square (hole): side 4 cm
What is the area of the colored part?
💡 Area with a hole
📐 Large square area:
10 × 10 = 100 cm²
📐 Hole area:
4 × 4 = 16 cm²
🔢 Colored area:
= 100 − 16 = 84 cm²
✨ Principle: Colored area = whole area − hole area
🎯 Rectangle with complex variable:
A rectangle has length \(2x+3\) and width \(x-1\).
Write a simplified expression for the perimeter..
💡 Expression with subtraction
📐 Perimeter:
= 2(2x + 3 + x − 1)
= 2(3x + 2)
= 6x + 4
🔢 Or the long way:
= 2(2x + 3) + 2(x − 1) = 4x + 6 + 2x − 2 = 6x + 4
🏗️ Composite shape with variable:
Two identical squares, side = \(x\).
a. What is the area of the shape?
b. If \(x=5\) what is the area?
💡 Area with powers
📐 Part a:
One square: x²
Two squares: x² + x² = 2x²
🔢 Part b:
x = 5 → 2×(5²) = 2×25 = 50 cm²
📐 Rectangle inside a square:
Square: side 8 cm
Inner rectangle: 6 cm × 4 cm
What is the area of the frame (the part between the square and the rectangle)?
💡 Frame area
📐 Outer square:
8 × 8 = 64 cm²
📐 Inner rectangle:
6 × 4 = 24 cm²
🔢 Frame area:
= 64 − 24 = 40 cm²
✨ Frame = outer area − inner area
🎨 Outer and inner perimeter:
Large rectangle: 20 cm × 12 cm
Inner rectangle (hole): 10 cm × 6 cm
What is the total perimeter (outer + inner)?
💡 Double perimeter
📐 Outer perimeter:
= 2×(20 + 12) = 64 cm
📐 Inner perimeter (hole):
= 2×(10 + 6) = 32 cm
🔢 Total perimeter:
= 64 + 32 = 96 cm
🔷 Staircase with variable:
3 stairs/steps. Each step: width \(x\), height \(x\).
Write an expression for the perimeter. the shape.
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💡 Stairs with a variable
🪜 Correct count:
📐 Bottom side: one long line = 3x
📐 Top (steps): 3 horizontal segments = 3x total
📐 Right side: one tall line = 3x
📐 Left steps: 3 vertical segments = 3x total
🔢 Total perimeter = 3x + 3x + 3x + 3x = 12x
🎯 Challenging problem:
Inverted T-shape:
Top part: 15 cm × 5 cm
Bottom part: 7 cm × 10 cm
What is the total area?
💡 T-shape
📐 Top part:
15 × 5 = 75 cm²
📐 Bottom part:
7 × 10 = 70 cm²
🔢 Total area:
= 75 + 70 = 145 cm²
🏆 Comprehensive summary:
Shape: square side \(2x\) with a square hole side \(x\).
a. Write an expression for the frame area.
b. Calculate the area when \(x=4\)
💡 Comprehensive question
📐 Part a — expression:
Outer square: (2x)² = 4x²
Inner square: x²
Frame area = 4x² − x² = 3x²
🔢 Part b — substitution:
x = 4 → 3×(4²) = 3×16 = 48 cm²