Q1.
What procedure would you use to square a given triangle? Here, the task is to construct a square whose area is equal to the area of some given triangle. Think carefully. How would you proceed?
Answer
Do it in two moves: triangle → rectangle → square. The second move is Baudhāyana's construction, which the section has just given us; only the first move is new, and it is easy.
Step 1 — turn the triangle into a rectangle of the same area.
- In ΔABC choose BC as base, of length b, and let the height from A be h.
- Mark the midpoints M of AB and N of AC and draw MN. By the midpoint theorem MN ∥ BC and MN = ½b, and MN is at height h/2.
- Drop perpendiculars from M and N to BC. The triangle is now cut into three pieces: a central rectangle and two right triangles, which fold up into the gaps. The result is a rectangle of base b and height h/2.
area of rectangle = b × h⁄2 = ½bh = area of the triangle ✓
Step 2 — square the rectangle. Apply Baudhāyana's construction of Fig. 6.30 to this rectangle with sides a = b and b′ = h/2: locate E, take the midpoint F of ED, build the square AFGH, swing the arc AG with centre H to meet BC at K, and so on. The square HPQS produced has area ab′, that is, ½bh.
side of the final square = √(½bh)
i.e. the geometric mean of the two sides of the rectangle
i.e. the geometric mean of the two sides of the rectangle
Why it happens: "squaring" a shape means finding the side s with s² = area, i.e. taking a square root by construction. Baudhāyana's trick is a geometric identity —
(a+b⁄2)² − (a−b⁄2)² = ab
— realised as a right triangle: HK = (a+b)/2 is the hypotenuse, BH = (a−b)/2 a leg, and the remaining leg HP satisfies HP² = ab by the Baudhāyana–Pythagoras theorem. Since every polygon can be cut into triangles, and every triangle can now be squared, and two squares can be combined into one by the Pythagoras theorem, every polygon can be squared.
(a+b⁄2)² − (a−b⁄2)² = ab
— realised as a right triangle: HK = (a+b)/2 is the hypotenuse, BH = (a−b)/2 a leg, and the remaining leg HP satisfies HP² = ab by the Baudhāyana–Pythagoras theorem. Since every polygon can be cut into triangles, and every triangle can now be squared, and two squares can be combined into one by the Pythagoras theorem, every polygon can be squared.
Did you know? Squaring the circle — the same task for a disc, with straightedge and compass alone — is impossible. That was settled in 1882, when Lindemann proved that π is not the root of any polynomial with whole-number coefficients.