1
Area(ABCD) = Area(ABEF)
2
Area(△ABC) = Area(△ABD)
3
Area(△ABE) = ½ × Area(ABCD)
4
Area(△ABC) = Area(△ABD) and common base AB ⇒ CD ∥ AB
FORMULA CHAPTER
Chapter 12 – Theorems on Area
WBBSEClass 09Mathematics4 formulas
Complete WBBSE Madhyamik Class 9 Theorems on Area list with equal-area relationships, conditions and proof-style worked examples for exam preparation.
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Parallelograms on the Same Base and Between the Same Parallels
Area(ABCD) = Area(ABEF)
SymbolMeaningUnit
ABCommon baselength units
hCommon perpendicular heightlength units
AreaArea of each parallelogramsquare units
Worked example
Question: Parallelograms ABCD and ABEF stand on the same base AB and lie between the same parallels. If Area(ABCD) = 48 cm², find Area(ABEF).
Calculation:
Step 1: Both parallelograms have the same base AB.
Step 2: They lie between the same parallels, so their perpendicular heights are equal.
Step 3: Therefore, Area(ABEF) = Area(ABCD) = 48 cm².
Answer: Area(ABEF) = 48 cm².
Triangles on the Same Base and Between the Same Parallels
Area(△ABC) = Area(△ABD)
SymbolMeaningUnit
ABCommon baselength units
C, DThird vertices on a line parallel to ABNot applicable
AreaArea of each trianglesquare units
Worked example
Question: Triangles ABC and ABD stand on the same base AB and lie between the same parallels. If Area(△ABC) = 30 cm², find Area(△ABD).
Calculation:
Step 1: Both triangles have common base AB.
Step 2: C and D lie on a line parallel to AB, so both triangles have the same height.
Step 3: Hence, Area(△ABD) = Area(△ABC) = 30 cm².
Answer: Area(△ABD) = 30 cm².
Triangle and Parallelogram on the Same Base
Area(△ABE) = ½ × Area(ABCD)
SymbolMeaningUnit
ABCommon baselength units
hCommon perpendicular heightlength units
Area(△ABE)Area of the trianglesquare units
Area(ABCD)Area of the parallelogramsquare units
Worked example
Question: Triangle ABE and parallelogram ABCD stand on the same base AB and lie between the same parallels. If Area(ABCD) = 64 cm², find Area(△ABE).
Calculation:
Step 1: The triangle and parallelogram have the same base and height.
Step 2: Area(△ABE) = ½ × Area(ABCD).
Step 3: Area(△ABE) = ½ × 64 = 32 cm².
Answer: Area(△ABE) = 32 cm².
Converse Equal-Area Triangle Theorem
Area(△ABC) = Area(△ABD) and common base AB ⇒ CD ∥ AB
SymbolMeaningUnit
ABCommon baselength units
C, DThird vertices of the trianglesNot applicable
CDLine joining the third verticeslength units
AreaArea of a trianglesquare units
Worked example
Question: Triangles ABC and ABD have common base AB and equal areas, with C and D on the same side of AB. Prove that CD is parallel to AB.
Calculation:
Step 1: Area(△ABC) = ½ × AB × height from C.
Step 2: Area(△ABD) = ½ × AB × height from D.
Step 3: Equal areas on common base AB imply equal perpendicular heights from C and D.
Step 4: Points at equal perpendicular distance from AB lie on a line parallel to AB.
Answer: CD ∥ AB.