194. The Parallelogram's Secret · base times height, every time

Try to break it

Drag D sideways to shear the parallelogram without changing its height — the area stays exactly the same. Now drag A or B to stretch the base; the area scales directly with the base length. Try to change the area without touching base or height — you can't.

How you build it

Construct a parallelogram with altitude.

• Draw segment AB — the base of the parallelogram. Keep it roughly horizontal.
• Mark point D above AB, slightly to the left or right of A. The offset gives the parallelogram its slant.
• Draw segment AD — the oblique side of the parallelogram.
• Through D, draw a line parallel to AB — this is the direction of the top side DC.
• Through B, draw a line parallel to AD — this is the other oblique side BC. The two parallel lines cross at C.
• Mark C at the intersection of the two parallel lines. You now have all four vertices of the parallelogram: A, B, C, D.
• Draw segment DC — the top side, parallel to AB.
• Draw segment BC — completing the parallelogram. Opposite sides AB ∥ DC and AD ∥ BC.
• Drop a perpendicular from D to AB — this is the altitude h. Area = AB × h, regardless of the slant of the sides.

The proof, step by step

Prove that the area of a parallelogram is base × height.

• Drop a perpendicular from D to AB, meeting at H.
• Cut the right triangle ADH and slide it to the right side of the parallelogram.
• The triangle fits exactly against BC, forming a rectangle ABCH'.
• The rectangle has the same base AB and height h as the original parallelogram.
• Since Area of rectangle = Base × height, the parallelogram's area is also Base × height.

Worked example

In parallelogram PQRS, side PQ is taken as the base. If PQ = 12 cm and the corresponding height between PQ and SR is 8 cm, find the area of the parallelogram.

Area of a parallelogram = Base × height. Here, Base = 12 cm and height = 8 cm. So, Area = 12 × 8 = 96 cm².

• 48 cm²
• 80 cm²
• 96 cm² — correct
• 192 cm²