159. Base Meets Height · Why area is always base times height

Try to break it

Drag A, B, or D to shear and stretch the parallelogram. The base and the perpendicular height change, but the area stays base × height. Slide D sideways without changing the height and watch the area stay locked. Try to make the area depend on the slant length — it never does.

How you build it

Construct a parallelogram with altitude.

• Mark point A — the left end of the base.
• Mark point B — the right end of the base.
• Draw segment AB — the base b.
• Mark point D anywhere above AB — the top-left corner of the parallelogram.
• Draw segment AD — the slant side.
• Click on B, then click on AD — this draws the line BC parallel to AD.
• Click on D, then click on AB — this draws the top side DC parallel to AB.
• Mark point C at the intersection of the two parallel lines — the parallelogram ABCD is complete.
• Drop a perpendicular from D to AB — this is the height h. Area = AB × h.

The proof, step by step

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

• Take parallelogram ABCD with base AB and height DE.
• Cut the right triangle ADE along the height DE.
• Move triangle ADE to the right side, attaching it to BC.
• The new shape is a rectangle with the same base and height.
• Since area of rectangle = base × height, the parallelogram has the same area.

Worked example

One side of a parallelogram is 20 cm and its distance from the opposite side is 16 cm. Find the area of the parallelogram.

Area = base × height = 20 cm × 16 cm = 320 cm².

• 320 cm² — correct
• 360 cm²
• 160 cm²
• 280 cm²