Mensuration
149. The Area Multiplier · How length units become area units
The area is always the square of the side length, regardless of size.
Unit conversion for area: when the linear unit changes by a factor k, the area unit changes by k². So 1 m² = 100² cm² = 10 000 cm² (not just 100 cm²). Always square the conversion factor for areas.
What this lesson covers
Try to break it
Drag P to resize the square. Double the side L and the area doesn't double — it quadruples. Try to find a square size where area = L instead of L² — you can't. Area scales with the square of the length.
How you build it
Construct a square of side 20.
- Select the square tool and draw a square.
- Adjust the size until the area reads 400 square units.
The proof, step by step
Prove that the area of a square scales with the square of its side length.
- Start with the length conversion: 1 m = 100 cm.
- Square both sides of the equation to find the area relationship.
- Result: 1 m² = 10,000 cm². The unit of area is the square of the unit of length.
Worked example
A square garden has a side length of 5 meters. What is the area of the garden in square centimeters?
Side = 5 m. Convert to cm: 5 × 100 = 500 cm. Area = side² = 500 × 500 = 250,000 cm². Remember: 1 m² = 10,000 cm², so 25 m² = 250,000 cm².
- 25 cm²
- 2,500 cm²
- 250,000 cm² — correct
- 2,500,000 cm²