150. The Rectangle's Boundary · always twice the sum of length and breadth

Try to break it

Drag C to stretch the rectangle long and thin, or short and wide. The length l and breadth b change, but the perimeter always equals 2(l + b). A very long thin rectangle has a huge perimeter — and the formula tracks it perfectly.

How you build it

Construct a rectangle.

• Mark point A anywhere on the canvas.
• Mark point B to the right of A — AB will be the length l.
• Draw segment AB — the base of the rectangle.
• Drop a perpendicular at A.
• Drop a perpendicular at B.
• Open compass to breadth b. Place at A and cut an arc on the perpendicular — this marks D.
• Mark point D — where the arc meets the perpendicular at A.
• With the same radius, place compass at B and cut an arc on the perpendicular — this marks C.
• Mark point C — where the arc meets the perpendicular at B.
• Join D and C — the rectangle ABCD is complete.

The proof, step by step

Prove that the perimeter of a rectangle is 2 × (length + breadth).

• A rectangle has four right angles and opposite sides are equal in length.
• The perimeter is the total distance around the shape.
• Perimeter = length + breadth + length + breadth.
• Perimeter = 2(length + breadth).

Worked example

A rectangular park measures 20 m in length and 15 m in breadth. Calculate the total length of the fence required to enclose the park.

Perimeter = 2(l + b) = 2(20 + 15) = 2(35) = 70 m.

• 35 m
• 70 m — correct
• 300 m
• 140 m