Areas of Parallelograms and Triangles
About 7 min · Mathematical Reasoning · English
Time worked out from the lesson's 565 words at 150 words a minute, plus a minute for each question.
This topic explores a beautiful geometric principle: when parallelograms or triangles share the same base and lie between the same parallel lines, their areas are equal. This concept is used in questions involving area comparisons, proof-based reasoning, and multi-step geometry problems. Mastery requires understanding why equal bases and equal heights guarantee equal areas, and how to apply this principle to complex figures where multiple shapes overlap or share boundaries.
Key ideas
- Equal bases, equal heights: If two parallelograms (or two triangles) have equal bases and equal heights, their areas are equal. The height is the perpendicular distance between the parallel lines containing the base and opposite side.
- Same base, same parallels: When two figures share the same base and their opposite vertices (or sides) lie on a line parallel to the base, they have the same height, hence equal areas.
- Parallelograms on the same base between same parallels have equal area: All parallelograms with base AB and opposite sides on a line parallel to AB have identical area = AB × h, where h is the distance between the parallels.
- Triangles on the same base between same parallels have equal area: All triangles with base AB and third vertex on a line parallel to AB have identical area = ½ × AB × h.
- A triangle is half a parallelogram: A triangle and a parallelogram on the same base and between the same parallels satisfy Area(triangle) = ½ Area(parallelogram).
Watch out for
- Confusing slant side with height: Students measure the slant side of a parallelogram instead of the perpendicular height. Fix: Always drop a perpendicular from the base to the opposite parallel line; that perpendicular distance is the height.
- Assuming "same base" means physically identical position: Two triangles can have "equal bases" (same length) at different locations but still have equal areas if their heights are equal. Fix: Focus on base length and height equality, not position.
- Forgetting the factor of ½ for triangles: When comparing a triangle and a parallelogram on the same base between same parallels, forgetting that triangle area = ½ parallelogram area. Fix: Always write ½ in the triangle formula.
Quick reference
- Same base + same parallels → equal areas for parallelograms or for triangles.
- Area of parallelogram = base × perpendicular height.
- Area of triangle = ½ base × perpendicular height.
- Triangle area = ½ parallelogram area (same base, same parallels).
- Median of a triangle splits it into two equal-area triangles.
Check yourself: 3 questions
1. A parallelogram ABCD and a triangle PCD lie on the same base CD and between the same parallel lines. If the area of parallelogram ABCD is 120 sq cm, what is the area of triangle PCD?
2. Two triangles ABC and DBC are on the same base BC and between the same parallel lines BC and AD. If BC = 12 cm and the perpendicular distance between the parallel lines is 8 cm, what is the sum of the areas of triangles ABC and DBC?
3. In the figure, ABCD is a parallelogram. P is a point on AB such that AP:PB = 2:3. Q is a point on CD such that CQ:QD = 2:3. If the area of parallelogram ABCD is 150 sq cm, what is the area of parallelogram APQD?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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