151. The Square's Secret Count · perimeter is always 4 times the side

Try to break it

Drag B to grow or shrink the square. The side s changes, and the perimeter P always reads exactly 4s — whether the square is tiny or huge. Try to find a square size where P ≠ 4s; you can't.

How you build it

Construct a square of side 150.

• Mark point A on the canvas.
• Mark point B to the right of A.
• Draw segment AB.
• At B, draw a line perpendicular to AB.
• With centre B and radius equal to AB, draw an arc to cut the perpendicular at C.
• Mark the point C where the arc meets the perpendicular.
• At A, draw a line perpendicular to AB.
• With centre A and the same radius AB, draw an arc to cut the perpendicular at D.
• Mark the point D where the arc meets the perpendicular.
• Draw segment CD, the top side of the square.

The proof, step by step

Prove that the perimeter of a square is 4 × side.

• A square is a quadrilateral with four equal sides and four right angles.
• Let the length of one side be denoted by 's'.
• Since all sides are equal, the other three sides also have length 's'.
• Perimeter is the total distance around the shape: P = s + s + s + s.
• Simplifying the sum gives P = 4s. Thus, the perimeter is four times the side length.

Worked example

The perimeter of a square park is 120 m. Find the length of its side.

Perimeter P = 4s. Given P = 120 m, we have 4s = 120. Dividing both sides by 4 gives s = 30 m.

• 20 m
• 30 m — correct
• 40 m
• 60 m