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Squaring a rectangle · 3 marks

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: Two stages: (1) turn the triangle into a rectangle of equal area: base b and height h2 (cut along the line joining the midpoints of two sides); (2) square that rectangle with Baudhāyana’s construction. The square has side √(bh2).

Step-by-step solution

Idea: We already know how to square a rectangle. So first make a rectangle with the same area as the triangle: area 12bh = b × h2.

  1. Triangle → rectangle. Mark the midpoint of the height (or join the midpoints of the two slant sides). Draw the rectangle on the base b with height h2. Its area is b × h2 = 12bh, the triangle’s area. (By cutting: the small top triangle above the midline splits along its height into two right triangles, which turn down to fill the two corners.)1 mark
  2. Rectangle → square. Apply Baudhāyana’s method to the rectangle with sides b and h2 (call the longer one p and the shorter q): mark E with AE = q, F the midpoint of ED, draw square AFGH, the arc from A with centre H meeting the rectangle’s far side at K, then KP ∥ AH; the square on HP has area pq.1 mark
  3. So the final square has area 12bh, equal to the triangle. Example: triangle with base 8 and height 9 has area 36; the rectangle is 8 × 4.5; its square has side 6.1 mark
Convert the triangle into a rectangle of base b and height h/2 (same area), then square that rectangle by Baudhāyana’s construction; the resulting square has the triangle’s area.

Check: Base 8, height 9: area 36. Rectangle 8 × 4.5 = 36. Square side 6, area 36 ✓.

Answer to write in the exam

ar(△) = 12bh = b × h2

Step 1: construct rectangle b × h2 (cut at mid-height)

Step 2: square this rectangle by Baudhāyana’s construction

∴ Square of side √(bh2) has the area of the triangle.

Common mistakes that cost marks

  • Using the rectangle b × h, which has twice the triangle’s area.
  • Taking the square’s side as b + h2; the side is √(area).

How this can come in the exam

MCQ (1 mark)

A triangle has base 12 cm and height 6 cm. The side of the square with the same area is

  1. 6 cm
  2. √72 cm
  3. 9 cm
  4. 36 cm
Show answer

(A) 6 cm
Area = 12 × 12 × 6 = 36 cm2, so the side is 6 cm.

Try one yourself

A triangle has base 10 cm and height 5 cm. What rectangle (on the same base) has the same area, and what is the side of the equal square?

Show answer

Rectangle 10 × 2.5 (area 25); square side 5 cm.

More questions like this

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← Try to work out why this method works. You will find that it is a geomFind the area of triangle ADE in the figure. →