Perimeter and perpendicular area
Separate boundary length from enclosed area for rectangles, triangles and parallelograms.
Section Paper 2 · Extended · Non-calculator
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Multiply lengths in matching units.
- Identify a perpendicular distance.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which unit measures area?
What you will learn
- Find rectangle perimeter and area.
- Use a perpendicular height in triangle and parallelogram areas.
Boundary and coverage answer different questions
Perimeter is the distance around a boundary, measured in length units. A rectangle with sides a and b has perimeter 2a + 2b and area ab. Area measures how much surface is enclosed and uses square units. A 9 cm by 4 cm rectangle has perimeter 26 cm and area 36 cm²; these values cannot be interchanged.
A triangle's perimeter is the sum of its three sides. Its area is half a base times the perpendicular height to that base, including when the height falls outside an obtuse triangle. A sloping side is not automatically the height. Choose a base, then identify the 90° distance from its line to the opposite vertex.
A parallelogram can be rearranged into a rectangle
Cutting a triangular corner from a parallelogram and moving it to the other end makes a rectangle with the same base b and perpendicular height h. Its area is therefore bh. This rearrangement changes the boundary but preserves area, so it cannot justify using the rectangle perimeter for the parallelogram.
For base 12 cm, sloping side 7 cm and perpendicular height 5 cm, a parallelogram has area 60 cm² and perimeter 38 cm. The side 7 determines part of the boundary; height 5 determines the area. If area and base are given instead, rearrange h = area/base, keeping square units divided by length units as length units.
Pause and explain
A triangle has base 10 cm and perpendicular height 6 cm. Find its area.
Worked example
A parallelogram has base 12 cm, side 7 cm and perpendicular height 5 cm. Find area and perimeter.
Show the worked solution
- Use perpendicular height for area: 12 × 5 = 60 cm².
- Use side lengths for perimeter: 2(12 + 7) = 38 cm.
- The two measurements answer different questions and have different units.
Answer Area 60 cm²; perimeter 38 cm.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the perimeter and area of an 8 cm by 3 cm rectangle.
Give me a hint
Calculate the boundary and coverage separately.
Compare my reasoning
- Perimeter is 2(8 + 3) = 22 cm.
- Area is 8 × 3 = 24 cm².
- Attach length units to perimeter and square units to area.
22 cm; 24 cm².
Look for these in your work
- Included both pairs of sides.
- Used distinct units.
A triangle has area 42 cm² and base 12 cm. Find its perpendicular height.
Give me a hint
Rearrange A = bh/2 before substituting.
Compare my reasoning
- Multiply both sides by two: 84 = 12h.
- Divide by 12 to get h = 7 cm.
- Check 12 × 7 ÷ 2 = 42 cm².
7 cm.
Look for these in your work
- Retained the factor one half.
- Verified the recovered height.
A parallelogram sign has base 1.8 m, side 1.2 m and perpendicular height 0.9 m. Find the paint area on one side and the edge trim length.
Give me a hint
Paint covers area; trim follows the boundary.
Compare my reasoning
- One-side area is 1.8 × 0.9 = 1.62 m².
- Trim length is 2(1.8 + 1.2) = 6 m.
- The sloping side is used for trim, not as the paint-area height.
1.62 m² of paint coverage; 6 m of trim.
Look for these in your work
- Used perpendicular height for area.
- Included every boundary side.
Common mistakes
- Using a sloping length as height.
- Giving area in centimetres.
- Forgetting to halve a triangle's base–height product.
What can you explain now?
Does a triangle's sloping side always equal its perpendicular height?
Compare with the explanation
No.
The height must meet the chosen base or its extension at a right angle.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, label the base, side and perpendicular height of a parallelogram, then calculate area and perimeter separately.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.