Give a method to convert a parallelogram into a rectangle of equal area.

You can try this using a cut-out of a parallelogram.

Question

Give a method to convert a parallelogram into a rectangle of equal area. You can try this using a cut-out of a parallelogram.

Homi · Accepted Answer

Understand the Question

A parallelogram can be converted into a rectangle of the same area by slicing off a right-angled triangle from one side along its altitude (height) and sliding it to the opposite side.
Since no area is added or removed during the cutting and rearranging process, the total area remains conserved: $\text{Area of Parallelogram} = \text{Area of Rectangle} = \text{base} \times \text{height}$

Step 1 · Draw the Altitude

Consider a parallelogram $\text{ABCD}$ . Draw a perpendicular line from vertex $\text{A}$ to the base $\text{DC}$ , meeting $\text{DC}$ at point $\text{X}$ . Diagram 2

$\text{AX}$ represents the height of the parallelogram.
$\text{DC}$ represents the base of the parallelogram.
$\Delta \text{ADX}$ forms a right-angled triangle.

Step 2 · Cut and Rearrange the Triangle

Cut out the right-angled triangle $\Delta \text{ADX}$ from the left side of the parallelogram. The remaining figure is the trapezoid $\text{ABCX}$ .
Shift the triangle $\Delta \text{ADX}$ to the right side of the trapezoid.
Align side $\text{AD}$ with the opposite side $\text{BC}$ (since $\text{AD} = \text{BC}$ ).

The newly formed figure is a rectangle with length equal to base $\text{DC}$ and breadth equal to height $\text{AX}$ .

Step 3 · Verify Equality of Area

The original parallelogram is made up of two parts: $\text{Area of parallelogram } \text{ABCD} = \text{Area of } \Delta \text{ADX} + \text{Area of trapezoid } \text{ABCX}$

The newly formed rectangle consists of the exact same two parts: $\text{Area of rectangle} = \text{length} \times \text{breadth} = \text{DC} \times \text{AX}$

Therefore: $\text{Area of parallelogram} = \text{Area of rectangle} = \text{base} \times \text{height}$

Answer

Cut the right-angled triangle along the height from one end of the parallelogram and attach it to the other end to form a rectangle of equal area ( $\text{base} \times \text{height}$ ).

Common Mistakes

Slant Height vs Perpendicular Height: Cutting along an arbitrary or slanted line instead of a perpendicular altitude will not produce the $90^\circ$ angles required to form a rectangle.
Incorrect Side Alignment: When shifting the cut triangle, its hypotenuse ( $\text{AD}$ ) must align with the parallel opposite side ( $\text{BC}$ ) so that the vertical altitude becomes the breadth of the rectangle.

Question 16

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