Area Theorems [Proof and Use]

254. Same Base, Same Parallels · Why a triangle is always half its parallelogram twin

Area(ΔABC) is always exactly half of Area(//gm ABDE).

ABEFArea(△ABC) = 60000Area(△ABC) = 60000Area(//gm ABDE) = 120000Area(//gm ABDE) = 120000CD
A triangle and a parallelogram on the same base and between the same parallels are linked by: the triangle is half the parallelogram. So Area(ΔABC) = ½ × Area(parallelogram ABDE).

Stuck? Ask Guru

Selina ICSE: Area Theorems [Proof and Use]

What this lesson covers

Try to break it

Drag C left or right along the top line to slide the triangle's apex. The height stays the same, so △ABC's area doesn't budge. Drag D to shear the parallelogram. The triangle's area is always exactly half the parallelogram's because they share the same base and the same parallels.

How you build it

Construct triangle ABC and parallelogram ABED on the same base AB, between the same parallels.

  • Point tool: mark A — the left end of the base.
  • Point tool: mark B — the right end of the base.
  • Segment tool: join A to B — the common base.
  • Parallel tool: click a spot above, then click AB — the top parallel that the triangle and parallelogram both reach.
  • Point tool: mark C anywhere on the top line — the triangle's apex.
  • Segment tool: join A to C.
  • Segment tool: join B to C — triangle ABC is complete.
  • Point tool: mark D elsewhere on the top line — the parallelogram's top-left corner.
  • Segment tool: join A to D — the left side of the parallelogram.
  • Parallel tool: click B, then click on AD — this is side BE; it meets the top line at E.
  • Point tool: mark E where the line through B meets the top line.
  • Segment tool: join B to E — parallelogram ABED is complete. Triangle ABC is exactly half its area (same base, same parallels).

The proof, step by step

Prove that a triangle is half the area of a parallelogram on the same base and between the same parallels.

  • BC is the diagonal of //gm ABFC, so Area(ΔABC) = ½ Area(//gm ABFC).
  • //gms ABFC and ABDE are on the same base AB and between the same parallels, so their areas are equal.
  • Therefore, Area(ΔABC) = ½ Area(//gm ABDE). Hence Proved.

Worked example

In the figure, ΔABC and parallelogram ABDE share the same base AB and lie between the same parallels. If the area of //gm ABDE is 120 cm², what is the area of ΔABC?

Since the triangle and parallelogram are on the same base and between the same parallels, Area(ΔABC) = ½ Area(//gm ABDE) = ½ × 120 = 60 cm².

  • 40 cm²
  • 60 cm² — correct
  • 80 cm²
  • 120 cm²
Hold to talk

Subscription Status