157. Square's Secret Area · Side squared or diagonal squared divided by 2

Try to break it

Drag H along the diagonal direction. The side s and diagonal d both grow, and the two area formulas — s² and d²/2 — always read the same number. Try to drag H to a setting where they disagree; you can't, because d² = 2s² in any square.

How you build it

Construct a square.

• Mark point A on the canvas.
• Mark point B to the right of A.
• Draw segment AB as the base of the square.
• At point A, draw a perpendicular to AB.
• With centre A and radius AB, draw a circle. It marks D on the perpendicular.
• Mark D where the circle meets the perpendicular at A.
• At point B, draw a perpendicular to AB.
• With centre B and the same radius AB, draw a circle. It marks C on the perpendicular.
• Mark C where the circle meets the perpendicular at B.
• Join C and D to complete the top side of the square.

The proof, step by step

Prove that the area of a square is side².

• A square has four equal sides and four right angles.
• Diagonal AC divides the square into two congruent right triangles: ABC and ADC.
• Area of triangle ABC = 1/2 × base × height = 1/2 × s × s.
• Area of square = 2 × Area of triangle ABC = 2 × (1/2 × s²) = s².
• Since diagonal d = s√2, we have s = d/√2.
• Area = s² = (d/√2)² = d²/2. Hence proved.

Worked example

The diagonal of a square measures 12 cm. Find the area of the square.

Area = 1/2 × d² = 1/2 × 12² = 1/2 × 144 = 72 cm².

• 72 cm² — correct
• 144 cm²
• 36 cm²
• 24 cm²