Area and Perimeter of Plane Figures
267. The Diagonal Split · Area of a quadrilateral in one formula
The area formula holds when B and D are on opposite sides of AC.
A diagonal splits a quadrilateral into two triangles, so its area = ½ × d × (h₁ + h₂), where d is the diagonal AC and h₁, h₂ are the perpendiculars to it from the opposite vertices B and D.
What this lesson covers
Try to break it
Drag B and D anywhere. The diagonal AC stays fixed; BX and DY are the perpendicular drops from B and D to AC. The quadrilateral's area always equals ½·AC·(BX + DY) — the diagonal times the sum of the two perpendiculars. Try to find a configuration where this fails; impossible.
How you build it
Construct a quadrilateral and split it along a diagonal to find its area.
- Point tool: mark A — left end of the diagonal.
- Point tool: mark C — right end of the diagonal.
- Segment tool: join A to C — the diagonal d.
- Point tool: mark B above the diagonal.
- Point tool: mark D below the diagonal — opposite side to B.
- Segment tool: join A to B.
- Segment tool: join B to C.
- Segment tool: join C to D.
- Segment tool: join D to A — quadrilateral ABCD is complete.
- Perp tool: click B, then click on AC — the perpendicular BX (height h₁).
- Point tool: mark X where BX meets AC.
- Perp tool: click D, then click on AC — the perpendicular DY (height h₂).
- Point tool: mark Y where DY meets AC. Area = ½ × AC × (BX + DY).
The proof, step by step
Prove that a diagonal splits a quadrilateral into two triangles whose areas sum to the whole.
- Area(ABCD) = Area(ABC) + Area(ADC)
- Area(ABC) = 1/2 * AC * BX
- Area(ADC) = 1/2 * AC * DY
- Area(ABCD) = 1/2 * AC * (BX + DY)
Worked example
In a quadrilateral ABCD, diagonal AC = 12 cm. Perpendiculars from B and D to AC are 5 cm and 7 cm respectively. Find the area of the quadrilateral.
Area = 1/2 * 12 * (5 + 7) = 6 * 12 = 72 cm².
- 60 cm²
- 72 cm² — correct
- 84 cm²
- 96 cm²