Areas of Parallelograms and Triangles
Concept-first revision with intuition, formal rules, worked examples, exam reasoning and focused practice.
Chapter Snapshot
- Parallelograms on the same base and between the same parallels have equal areas.
- Triangles on the same base and between the same parallels have equal areas.
- A triangle and parallelogram on the same base and between the same parallels have areas in the ratio 1:2.
- Equal area does not imply congruence.
- Area reasoning often uses a common altitude rather than side-by-side measurement.
What You Will Learn
- Understand figures on the same base and between the same parallels
- Compare areas without measuring every side
- Use area relationships for parallelograms
- Use area relationships for triangles
- Relate triangle and parallelogram areas
- Solve proof-style area problems
Why This Chapter Matters
This chapter changes how you think about area. Earlier, area meant substituting lengths into formulas. Here, the emphasis is comparison: two figures can have equal areas even when their shapes look very different.
The key idea is same base, same height. If two shapes lie between the same parallel lines, their perpendicular distance—their height—is the same. Once the base is also shared, their areas become directly comparable.
The chapter is therefore less about arithmetic and more about seeing hidden equal heights created by parallel lines.
Bridge from Earlier Classes
Before starting, be comfortable with:
- Area of triangle
- Area of parallelogram
- Parallel lines
- Perpendicular distance
These are the tools this chapter reuses and deepens.
1. Same Base and Same Parallels
Two figures are on the same base when they share the same line segment as a base. They are between the same parallels when their opposite vertices or sides lie on a line parallel to that base.
Parallel lines remain at a constant perpendicular distance, so they create a common height. That shared height drives all the main theorems in this chapter.
2. Parallelograms on the Same Base
If two parallelograms stand on the same base and lie between the same parallels, they have the same base and the same perpendicular height.
Since area = base × height, their areas are equal even if one appears more slanted than the other.
3. Triangles on the Same Base
Two triangles on the same base and between the same parallels also have the same height.
Because triangle area is half base times height, equal base and equal height produce equal area. Shape can change while area remains unchanged.
4. Triangle vs Parallelogram
If a triangle and parallelogram share the same base and lie between the same parallels, they share the same height.
The triangle area is 1/2 bh while the parallelogram area is bh. So the triangle has exactly half the parallelogram's area.
5. Equal Area Is Not Congruence
Equal-area figures need not have the same shape or matching sides. A long slanted parallelogram and a less slanted one can enclose the same area if their base and perpendicular height are equal.
Congruence concerns full shape and size; equal area concerns only the amount of surface enclosed.
6. Proof Problems
Many questions ask you to prove areas equal by identifying a common base or common altitude.
A reliable strategy is to write both area expressions with the same base or height and then cancel the common factors. Avoid judging area by appearance.
Visualise the Core Idea
Parallel lines create a common height, which drives the area relationships.
Formula / Theorem Sheet
Formula / Key Relations
Parallelogram area
b is base, h is perpendicular height.
Triangle area
Use perpendicular height.
Same base/same parallels
When base and height are shared.
Worked Examples — Follow the Reasoning
They have the same base AB and equal perpendicular height.
Therefore their areas are equal.
How to Think in the Exam
Look for a shared base and common height created by parallel lines before calculating. In proof questions, area formulas are often used comparatively rather than numerically.
Common Mistakes
Key Terms
Quick Revision
Quick Revision
- Parallelograms on the same base and between the same parallels have equal areas.
- Triangles on the same base and between the same parallels have equal areas.
- A triangle and parallelogram on the same base and between the same parallels have areas in the ratio 1:2.
- Equal area does not imply congruence.
- Area reasoning often uses a common altitude rather than side-by-side measurement.
- Revise the reasoning behind each rule once, not only the final formula.
Focused Practice
Two triangles on the same base and between the same parallels have what relation?
View Solution
Equal areas.
A parallelogram area is 96 cm². Find area of a triangle on same base and parallels.
View Solution
48 cm²
Does equal area imply congruence?
View Solution
No.
Why can a slanted parallelogram have the same area as a less slanted one?
View Solution
If base and perpendicular height stay unchanged, bh stays unchanged.
What is the central idea behind equal-area theorems in this chapter?
View Solution
Shared base and equal perpendicular height.
Final Chapter Summary
Chapter Summary
- Parallelograms on the same base and between the same parallels have equal areas.
- Triangles on the same base and between the same parallels have equal areas.
- A triangle and parallelogram on the same base and between the same parallels have areas in the ratio 1:2.
- Equal area does not imply congruence.
- Area reasoning often uses a common altitude rather than side-by-side measurement.
- Strong revision means understanding why the method works, then practising enough examples to recognise when to use it.