A rectangle (8 units × 5 units) with a trapezoid (bases 6 units and 4 units, height 3 units) attached to its top side.
Step 1: Break the composite shape into two parts: rectangle and trapezoid.
Step 2: Calculate the area of the rectangle: 8 × 5 = 40 square units.
Step 3: Calculate the area of the trapezoid: ½ × (6 + 4) × 3 = ½ × 10 × 3 = 15 square units.
Step 4: Since the shapes are joined together, add their areas: 40 + 15 = 55 square units.
Trapezoid: A quadrilateral with one pair of parallel sides called bases.
- Break the composite shape into simpler shapes (rectangle and trapezoid)
- Calculate the area of each simple shape
- Add all the areas together
Shape consists of: Rectangle (8×5) and Trapezoid (bases 6,4; height 3)
Rectangle area: 8 × 5 = 40 square units
Trapezoid area: ½ × (6 + 4) × 3 = ½ × 10 × 3 = 15 square units
Total area = 40 + 15 = 55 square units
The area of the composite shape is 55 square units.
• Area of rectangle: length × width
• Area of trapezoid: ½ × (sum of bases) × height
• Composite area: Sum of areas of all parts
A triangle (base 6 units, height 4 units) attached to the top of a rectangle (6 units × 3 units).
Step 1: Identify the shapes: triangle and rectangle.
Step 2: Calculate the area of the triangle: ½ × 6 × 4 = 12 square units.
Step 3: Calculate the area of the rectangle: 6 × 3 = 18 square units.
Step 4: Since the shapes are joined together, add their areas: 12 + 18 = 30 square units.
Triangle: A three-sided polygon with three angles.
Shape consists of: Triangle (base 6, height 4) and Rectangle (6×3)
Triangle area: ½ × base × height = ½ × 6 × 4 = 12 square units
Rectangle area: length × width = 6 × 3 = 18 square units
Total area = 12 + 18 = 30 square units
The area of the composite shape is 30 square units.
• Area of triangle: ½ × base × height
• Area of rectangle: length × width
• Composite area: Sum of areas of all parts
A trapezoid (bases 5 units and 3 units, height 4 units) with a triangle (base 5 units, height 2 units) on top.
Step 1: Identify the shapes: trapezoid and triangle.
Step 2: Calculate the area of the trapezoid: ½ × (5 + 3) × 4 = ½ × 8 × 4 = 16 square units.
Step 3: Calculate the area of the triangle: ½ × 5 × 2 = 5 square units.
Step 4: Since the shapes are joined together, add their areas: 16 + 5 = 21 square units.
Composite shape with trapezoid and triangle: A shape made up of a trapezoid and a triangle joined together.
Shape consists of: Trapezoid (bases 5,3; height 4) and Triangle (base 5, height 2)
Trapezoid area: ½ × (sum of bases) × height = ½ × (5 + 3) × 4 = 16 square units
Triangle area: ½ × base × height = ½ × 5 × 2 = 5 square units
Total area = 16 + 5 = 21 square units
The area of the composite shape is 21 square units.
• Area of trapezoid: ½ × (sum of bases) × height
• Area of triangle: ½ × base × height
• Composite area: Sum of areas of all parts
A rectangle (7 units × 4 units) with a triangle (base 7 units, height 3 units) on top and a trapezoid (bases 7 units and 5 units, height 2 units) attached to the right side.
Step 1: Identify the shapes: rectangle, triangle, and trapezoid.
Step 2: Calculate the area of the rectangle: 7 × 4 = 28 square units.
Step 3: Calculate the area of the triangle: ½ × 7 × 3 = 10.5 square units.
Step 4: Calculate the area of the trapezoid: ½ × (7 + 5) × 2 = ½ × 12 × 2 = 12 square units.
Step 5: Since the shapes are joined together, add their areas: 28 + 10.5 + 12 = 50.5 square units.
Complex composite shape: A shape made up of multiple simple shapes including rectangles, triangles, and trapezoids arranged in different positions.
Shape consists of: Rectangle (7×4), Triangle (base 7, height 3), Trapezoid (bases 7,5; height 2)
Rectangle area: 7 × 4 = 28 square units
Triangle area: ½ × base × height = ½ × 7 × 3 = 10.5 square units
Trapezoid area: ½ × (sum of bases) × height = ½ × (7 + 5) × 2 = 12 square units
Total area = 28 + 10.5 + 12 = 50.5 square units
The area of the complex composite shape is 50.5 square units.
• Area of rectangle: length × width
• Area of triangle: ½ × base × height
• Area of trapezoid: ½ × (sum of bases) × height
• Composite area: Sum of areas of all parts
An L-shaped figure made of a 6×4 rectangle with a 2×2 square removed from the bottom right corner, plus a triangle (base 4 units, height 3 units) attached to the bottom.
Step 1: Calculate the area of the L-shape: Full rectangle (6×4) minus removed square (2×2) = 24 - 4 = 20 square units.
Step 2: Calculate the area of the triangle: ½ × 4 × 3 = 6 square units.
Step 3: Since the shapes are joined together, add their areas: 20 + 6 = 26 square units.
L-shaped figure: A composite shape formed by removing one rectangle from another, creating an L pattern, sometimes with additional shapes attached.
L-shape area = Full rectangle - Removed square = (6 × 4) - (2 × 2) = 24 - 4 = 20 square units
Triangle area: ½ × base × height = ½ × 4 × 3 = 6 square units
Total area = L-shape area + Triangle area = 20 + 6 = 26 square units
The area of the L-shaped figure with triangle is 26 square units.
• Area of rectangle: length × width
• Area of square: side × side
• Area of triangle: ½ × base × height
• L-shape area: Full area - Removed area
• Composite area: Sum of areas of all parts
Trapezoid: A quadrilateral with exactly one pair of parallel sides (called bases).
Triangle: A three-sided polygon with three angles that sum to 180°.
The parallel sides of a trapezoid are always the bases, regardless of their orientation.
Don't confuse the height of a triangle with one of its sides!
Definitions:
Composite Shape: A shape made up of two or more basic geometric shapes like rectangles, squares, triangles, trapezoids, or circles.
Area: The amount of space inside a 2D shape, measured in square units.
Decomposition: Breaking a complex shape into simpler shapes whose areas we can calculate.
Trapezoid: A quadrilateral with exactly one pair of parallel sides (bases).
Triangle: A three-sided polygon with three angles that sum to 180°.
Core Rules and Laws:
- Area Addition Rule: When shapes don't overlap, add their areas together.
- Area Subtraction Rule: When one shape is cut out of another, subtract the cut-out area.
- Area Overlap Rule: When shapes overlap, add all areas and subtract the overlapping area.
- Rectangle Area Formula: Area = length × width
- Square Area Formula: Area = side × side
- Triangle Area Formula: Area = ½ × base × height
- Trapezoid Area Formula: Area = ½ × (sum of bases) × height
Step-by-Step Method:
- Look at the shape: Identify the different parts that make up the composite shape
- Label the parts: Give each simple shape a number or name
- Measure dimensions: Find the length and width of each simple shape
- Calculate each area: Use the correct formula for each shape
- Determine the relationship: Decide if you add, subtract, or account for overlaps
- Add the areas: Sum all the individual areas together following the relationship
Examples:
Simple Example: A 3×4 rectangle attached to a 2×2 square.
Rectangle area: 3 × 4 = 12 square units
Square area: 2 × 2 = 4 square units
Total area: 12 + 4 = 16 square units
Advanced Example: A 6×6 square with a 2×3 rectangle removed from one corner.
Full square area: 6 × 6 = 36 square units
Removed rectangle area: 2 × 3 = 6 square units
Remaining area: 36 - 6 = 30 square units
Tips and Tricks:
- Tip 1: Draw lines to separate the composite shape into rectangles, squares, triangles, or trapezoids.
- Tip 2: Write down the area of each simple shape before adding them together.
- Tip 3: Count the unit squares in a grid if you're not sure about your calculation.
- Tip 4: Always include the word "square units" in your final answer.
- Tip 5: Check your work by estimating if the answer makes sense for the size of the shape.
Common Pitfalls:
- Pitfall 1: Forgetting to account for overlapping areas
- Pitfall 2: Adding instead of subtracting when a part is removed
- Pitfall 3: Miscalculating dimensions of the parts
- Pitfall 4: Confusing area with perimeter
- Pitfall 5: Not using consistent units throughout
Memory Aids:
- Memory Aid 1: "A for Area" - Area is always measured in square units
- Memory Aid 2: "LW for Rectangle" - Length × Width
- Memory Aid 3: "SS for Square" - Side × Side
- Memory Aid 4: "BH divided by 2 for Triangle" - Base × Height ÷ 2
- Memory Aid 5: "Average of bases times height for Trapezoid" - ½(b₁ + b₂)h
- Memory Aid 6: "Add when joining, subtract when cutting" - Remember the relationship between parts
Key Takeaways:
- Every complex shape can be broken down into simpler shapes
- The total area is found by adding or subtracting the areas of the parts
- Always check your work by verifying that the answer makes sense
- Practice with different types of composite shapes to build confidence
- Pay attention to whether shapes overlap or if parts are removed
Composite shape: A shape made up of two or more basic geometric shapes like rectangles, squares, triangles, trapezoids, or circles.
Area: The amount of space inside a 2D shape, measured in square units.
Decomposition: Breaking a complex shape into simpler shapes whose areas we can calculate.
- Look at the shape: Identify the different parts that make up the composite shape
- Label the parts: Give each simple shape a number or name
- Measure dimensions: Find the length and width of each simple shape
- Calculate each area: Use the correct formula for each shape
- Add the areas: Sum all the individual areas together
• Area of rectangle: A = length × width
• Area of square: A = side × side (or side²)
• Area of triangle: A = ½ × base × height
• Area of trapezoid: A = ½ × (base₁ + base₂) × height
• Area of composite shape: Add all individual areas
• When shapes overlap: Add areas and subtract the overlap
Questions & Answers
Question: I don't understand how to identify the bases of a trapezoid. How do I know which sides are the bases when calculating the area?
Answer: Great question! Here's how to identify the bases of a trapezoid:
- Look for parallel sides: The bases of a trapezoid are the two sides that are parallel to each other.
- They may be different lengths: One base is usually longer than the other.
- Orientation doesn't matter: The parallel sides could be horizontal, vertical, or slanted.
- Only one pair: A trapezoid has exactly one pair of parallel sides (the bases) and one pair of non-parallel sides (the legs).
For example, if you see a four-sided shape where the top and bottom sides are parallel (even if they're different lengths), those are the bases. The left and right sides would be the legs.
Remember: Area of trapezoid = ½ × (base₁ + base₂) × height
Question: How do I find the height of a triangle when it's not given directly? What if the triangle is tilted?
Answer: The height of a triangle is always the perpendicular distance from the base to the opposite vertex. Here's how to find it:
- Look for the right angle: The height forms a 90° angle with the base.
- It might extend outside: For obtuse triangles, the height line may extend beyond the triangle.
- Choose any side as base: You can pick any side to be the base, then measure the height to the opposite vertex.
- Use given measurements: Often, the height is shown as a dashed line with a right angle symbol.
Even if a triangle is tilted, the height is still measured perpendicularly from the base to the highest point. The formula remains: Area = ½ × base × height.
Question: When I calculate the area of a composite shape with a trapezoid and triangle, should I calculate each shape separately and then add them?
Answer: Yes, that's exactly the correct approach! Here's the method:
- Separate the shapes: Identify and label each simple shape in the composite figure.
- Calculate each area: Use the correct formula for each shape:
- Rectangle: A = length × width
- Triangle: A = ½ × base × height
- Trapezoid: A = ½ × (base₁ + base₂) × height
- Add the areas: If the shapes are joined together, add all the individual areas.
- Subtract if needed: If one shape is cut out of another, subtract that area.
For example, if you have a rectangle and a triangle joined together, calculate: Area of rectangle + Area of triangle = Total area.
Always check that your answer makes sense for the size of the composite shape!