Perimeter and Area of Plane Figures
81. The Square's Secret · Why area is always side squared
The area of the square is always the square of its side length.
Area of a square = side × side = a². Since all four sides of a square are equal, its area is the square of its side length. Drag the corner — area = a² updates live.
What this lesson covers
Try to break it
Try doubling the side length. What happens to the area? It becomes four times bigger!
How you build it
Construct a square of side 15.
- Place point A, the first corner of the square.
- Place point B. The segment AB will be the first side of the square.
- Draw the segment from A to B.
- Construct a line through A, perpendicular to AB — this carries the next side.
- Set the compass opening to AB — click A, then B. The arc from A crosses the perpendicular.
- Mark point D where the arc crosses the perpendicular, above AB. Now AD = AB.
- With the compass still set to AB, draw an arc centred at B — click B.
- With the same opening, draw an arc centred at D — it crosses the arc from B.
- Mark point C where the arcs from B and D cross — the fourth corner.
- Draw the segment from B to C.
- Draw the segment from D to C. All four sides are equal and meet at right angles — ABCD is a square.
The proof, step by step
Why is the area of a square always side squared? Let's break it down step by step and see the logic behind the formula.
- A square is a special rectangle where all four sides are equal.
- The area of a rectangle is calculated as length × breadth.
- In a square, length = breadth = s.
- Substituting these values, Area = s × s = s².
Worked example
A square-shaped tablecloth has a side length of 1.5 meters. What is the area of the tablecloth?
Area = side² = 1.5² = 2.25 m².
- 3 m²
- 2.25 m² — correct
- 6 m²
- 1.5 m²