269. Area & Perimeter: Rect & Square · Master the formulas by dragging and exploring

Try to break it

Drag l and b on the rectangle, and a on the square. Every angle stays 90° as you go — the constraints lock the corners. The rectangle's readouts follow area = l·b and perimeter = 2(l + b); the square's follow area = a² and perimeter = 4a. Try to get any readout to break its formula; impossible.

How you build it

Build each figure from scratch with ruler and compass.

• Point tool: mark A — the left end of the base.
• Point tool: mark B to the right of A — AB is the length l.
• Segment tool: join A to B — the length l.
• Perp tool: click A, then click upward — a line square to AB at A.
• Perp tool: click B, then click upward — a line square to AB at B.
• Arc tool: click A, then click out to the breadth you want — the arc crosses A's perpendicular (this width is breadth b).
• Point tool: mark D where the arc cuts A's perpendicular.
• Arc tool: centre B, same width — the arc crosses B's perpendicular at the same height.
• Point tool: mark C where the arc cuts B's perpendicular.
• Segment tool: join D to C. ABCD is a rectangle — right angles at A and B, and AD = BC. Area = l × b.
• Point tool: mark P — the left end of the square's base.
• Point tool: mark Q to the right of P — PQ is the side a.
• Segment tool: join P to Q — the side a.
• Perp tool: click P, then click upward — square to PQ at P.
• Perp tool: click Q, then click upward — square to PQ at Q.
• Arc tool: click P, then Q — the radius equals side a. The arc crosses P's perpendicular at height a.
• Point tool: mark S where the arc cuts P's perpendicular — PS = a.
• Arc tool: centre Q, same radius a — the arc crosses Q's perpendicular.
• Point tool: mark R where the arc cuts Q's perpendicular — QR = a.
• Segment tool: join S to R. PQRS is a square — four equal sides a and four right angles. Area = a².

The proof, step by step

Prove that a rectangle has area l × b and a square has area a².

• Consider a rectangle with length l and breadth b.
• Divide the rectangle into unit squares of side 1.
• There are l squares along the length and b squares along the breadth.
• Total number of unit squares = l × b.
• Therefore, Area = l × b.
• For a square, l = b = a.
• Therefore, Area = a × a = a².

Worked example

A rectangular garden has a length of 15 m and a breadth of 8 m. Find its area and perimeter.

Area = l × b = 15 × 8 = 120 m². Perimeter = 2(l + b) = 2(15 + 8) = 2(23) = 46 m.

• Area = 120 m², Perimeter = 46 m — correct
• Area = 120 m², Perimeter = 23 m
• Area = 23 m², Perimeter = 46 m
• Area = 120 m², Perimeter = 92 m