Year 7 Mathematics · AC9M7M01

Area of Triangles and Parallelograms

We are learning to solve problems involving the area of triangles and parallelograms using established formulas and appropriate units

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Learning goalsSay it simply

We are learning to solve problems involving the area of triangles and parallelograms using established formulas and appropriate units.

Triangle and parallelogram area depends on a chosen base and its perpendicular height, not on an arbitrary sloping side. Area is measured in square units.

A parallelogram has area bh; rearranging or duplicating a triangle shows that its area is half the matching parallelogram, A=½bh.

For composite regions, partition or subtract non-overlapping pieces with compatible dimensions. Verify by using a second decomposition that covers the same region exactly once.

Success criteria

  • I can identify a valid base and perpendicular height.
  • I can calculate triangle and parallelogram areas with square units.
  • I can solve and independently check a composite-area problem.
Key conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Find a triangle area

  1. Identify base 12 cm and perpendicular height 7 cm.
  2. Use A=½bh.
  3. Calculate ½×12×7.
  4. Attach square units.

Final answer: 42 cm²

Check: Two copies form a 12×7 parallelogram of area 84 cm².

Example 2

Find a composite area

  1. Start with a 10×8 rectangle.
  2. Find its area as 80 cm².
  3. Subtract the 3×4 cut-out of 12 cm².
  4. State the remaining area.

Final answer: 68 cm²

Check: Add non-overlapping rectangles 7×8 and 3×4 to get 68 cm².

Example 3

Application problem 1

Problem: A parallelogram has base 12 cm and perpendicular height 7 cm. A triangle with base 8 cm and height 5 cm is cut from it. Find the remaining area and state the correct units.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: Parallelogram area = 84 cm²; triangle area = 20 cm²; remaining area = 64 cm².
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Final answer: Parallelogram area = 84 cm²; triangle area = 20 cm²; remaining area = 64 cm².

Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Application problem 2

Problem: A parallelogram and triangle share base 14 cm and height 5 cm. Compare areas.

  1. Plan: Apply both formulas to the same base-height pair.
  2. Work: The parallelogram area is 14×5=70 cm² and the triangle area is ½×14×5=35 cm², so their areas compare in the ratio 2:1.
  3. Interpret: The complete model for “A parallelogram and triangle share base 14 cm and height 5 cm. Compare areas.” shows The parallelogram area is 14×5=70 cm² and the triangle area is ½×14×5=35 cm², so their areas compare in the ratio 2:1.

Final answer: The parallelogram area is 14×5=70 cm² and the triangle area is ½×14×5=35 cm², so their areas compare in the ratio 2:1.

Check: The complete model for “A parallelogram and triangle share base 14 cm and height 5 cm. Compare areas.” shows The parallelogram area is 14×5=70 cm² and the triangle area is ½×14×5=35 cm², so their areas compare in the ratio 2:1.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7M01 - Area of Triangles and Parallelograms
Example 1

Example 1 Find a triangle area Identify base 12 cm and perpendicular height 7 cm. Use A=½bh. Calculate ½×12×7. Attach square units. Final answer: 42 cm² Check: Two copies form a 12×7 parallelogram of area 84 cm².

Example 2

Example 2 Find a composite area Start with a 10×8 rectangle. Find its area as 80 cm². Subtract the 3×4 cut-out of 12 cm². State the remaining area. Final answer: 68 cm² Check: Add non-overlapping rectangles 7×8 and 3×4 to get 68 cm².

Example 3

Example 3 Application problem 1 Problem: A parallelogram has base 12 cm and perpendicular height 7 cm. A triangle with base 8 cm and height 5 cm is cut from it. Find the remaining area and state the correct units. Plan: Represent the information first, then calculate, interpret and independently check the result. Work: Parallelogram area = 84 cm²; triangle area = 20 cm²; remaining area = 64 cm². Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. Final answer: Parallelogram area = 84 cm²; triangle area = 20 cm²; remaining area = 64 cm². Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Example 4 Application problem 2 Problem: A parallelogram and triangle share base 14 cm and height 5 cm. Compare areas. Plan: Apply both formulas to the same base-height pair. Work: The parallelogram area is 14×5=70 cm² and the triangle area is ½×14×5=35 cm², so their areas compare in the ratio 2:1. Interpret: The complete model for “A parallelogram and triangle share base 14 cm and height 5 cm. Compare areas.” shows The parallelogram area is 14×5=70 cm² and the triangle area is ½×14×5=35 cm², so their areas compare in the ratio 2:1. Final answer: The parallelogram area is 14×5=70 cm² and the triangle area is ½×14×5=35 cm², so their areas compare in the ratio 2:1. Check: The complete model for “A parallelogram and triangle share base 14 cm and height 5 cm. Compare areas.” shows The parallelogram area is 14×5=70 cm² and the triangle area is ½×14×5=35 cm², so their areas compare in the ratio 2:1.

Curriculum examplesCopied content

Content description: solve problems involving the area of triangles and parallelograms using established formulas and appropriate units.

Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. Find the area of a parallelogram with base 9 cm and height 4 cm.

    Check answer

    Answer: 36 cm².

    Hint: Multiply base by perpendicular height.

    Why: Using the method “Multiply base by perpendicular height.” gives 36 cm².

  2. 2. Find the area of a triangle with base 10 m and height 6 m.

    Check answer

    Answer: 30 m².

    Hint: Use one-half times base times height.

    Why: Using the method “Use one-half times base times height.” gives 30 m².

  3. 3. Which unit measures area when lengths are metres?

    Check answer

    Answer: Square metres, m².

    Hint: Identify area as a two-dimensional measure, then choose square units.

    Why: Using the method “Area is two-dimensional.” gives Square metres, m².

  4. 4. A triangle has area 24 cm² and base 8 cm. Find its height.

    Check answer

    Answer: 6 cm.

    Hint: Use h=2A÷b.

    Why: Using the method “Use h=2A÷b.” gives 6 cm.

  5. 5. Find the area of a parallelogram with base 7.5 m and height 3 m.

    Check answer

    Answer: 22.5 m².

    Hint: Use the perpendicular height.

    Why: Using the method “Use the perpendicular height.” gives 22.5 m².

  6. 6. A 12×9 rectangle loses a 4×3 corner. Find remaining area.

    Check answer

    Answer: 96 square units.

    Hint: Subtract the cut-out once.

    Why: Using the method “Subtract the cut-out once.” gives 96 square units.

  7. 7. Why is a 6 cm sloping side not automatically a triangle height?

    Check answer

    Answer: A triangle height is the perpendicular distance to the chosen base; a 6 cm sloping side is a valid height only if it meets that base at 90°.

    Hint: Look for a right-angle relationship.

    Why: The reasoning for “Why is a 6 cm sloping side not automatically a triangle height?” is complete because A triangle height is the perpendicular distance to the chosen base; a 6 cm sloping side is a valid height only if it meets that base at 90°.

  8. 8. A triangular garden has area 54 m² and height 9 m. Find its base.

    Check answer

    Answer: From 54=½×b×9, multiply by 2 and divide by 9 to get b=108÷9=12 m; checking gives ½×12×9=54 m².

    Hint: Rearrange b=2A÷h.

    Why: The complete model for “A triangular garden has area 54 m² and height 9 m. Find its base.” shows From 54=½×b×9, multiply by 2 and divide by 9 to get b=108÷9=12 m; checking gives ½×12×9=54 m².

  9. 9. A parallelogram and triangle share base 14 cm and height 5 cm. Compare areas.

    Check answer

    Answer: The parallelogram area is 14×5=70 cm² and the triangle area is ½×14×5=35 cm², so their areas compare in the ratio 2:1.

    Hint: Apply both formulas to the same base-height pair.

    Why: The complete model for “A parallelogram and triangle share base 14 cm and height 5 cm. Compare areas.” shows The parallelogram area is 14×5=70 cm² and the triangle area is ½×14×5=35 cm², so their areas compare in the ratio 2:1.

  10. 10. A parallelogram has base 12 cm and perpendicular height 7 cm. A triangle with base 8 cm and height 5 cm is cut from it. Find the remaining area and state the correct units.

    Check answer

    Answer: Parallelogram area = 84 cm²; triangle area = 20 cm²; remaining area = 64 cm².

    Hint: Represent the information first, then calculate, interpret and independently check the result.

    Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Practice and reviewReady for practice

Common mistake: Sloping side used as height.

Correction: Locate or construct the perpendicular distance.

Common mistake: Triangle factor one-half omitted.

Correction: Two congruent triangles form a parallelogram.

Common mistake: Mixed length units multiplied.

Correction: Convert first.

Curriculum alignmentStart here

We are learning to solve problems involving the area of triangles and parallelograms using established formulas and appropriate units.

Triangle and parallelogram area depends on a chosen base and its perpendicular height, not on an arbitrary sloping side. Area is measured in square units.

A parallelogram has area bh; rearranging or duplicating a triangle shows that its area is half the matching parallelogram, A=½bh.

For composite regions, partition or subtract non-overlapping pieces with compatible dimensions. Verify by using a second decomposition that covers the same region exactly once.

Success criteria

  • I can identify a valid base and perpendicular height.
  • I can calculate triangle and parallelogram areas with square units.
  • I can solve and independently check a composite-area problem.
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