How can one construct a square having double the area of a given square?

Question

Homi · Accepted Answer

We can use the diagonal of the original square to find the side length of the new square.

Step 1 — Understand the Area Relationship

Let the side length of the given square be $s$ . The area of this square is found by multiplying its side by itself.

$\text{Area of original square} = s \times s$

$= s^2$

We want to construct a new square that has double this area. So, the area of the new square must be $2 \times s^2$ . Let the side length of this new square be $S$ . The area of the new square is $S \times S$ , which is $S^2$ .

$S^2 = 2s^2$

To find the side length $S$ , we take the square root of both sides.

$S = \sqrt{2s^2}$

$S = s\sqrt{2}$

$\boxed{\text{The new side length is } s\sqrt{2}}$

This means the side of the new square must be $\sqrt{2}$ times the side of the original square.

Diagram 1

Step 2 — Find the Length $s\sqrt{2}$

Let us consider the given square, with vertices A, B, C, D. Draw a diagonal of this square, for example, the line segment AC. The diagonal AC divides the square into two right-angled triangles. Let's look at triangle ABC. It has a right angle at B. The sides AB and BC are the legs of this right-angled triangle. Both AB and BC have length $s$ . The diagonal AC is the hypotenuse. We can use the Pythagorean theorem to find the length of AC. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

$AC^2 = AB^2 + BC^2$

$AC^2 = s^2 + s^2$

$AC^2 = 2s^2$

To find AC, we take the square root.

$AC = \sqrt{2s^2}$

$AC = s\sqrt{2}$

$\boxed{\text{The diagonal AC has length } s\sqrt{2}}$

So, the diagonal of the original square has the exact length we need for the side of our new square.

Diagram 2

Step 3 — Construct the New Square

Now we will construct the new square using the diagonal AC as one of its sides.

Draw the original square, ABCD.
Draw one of its diagonals, for example, AC. This diagonal will be the side of our new square.
At point A, construct a line perpendicular to AC. You can do this using a compass and straightedge.
At point C, construct a line perpendicular to AC.
Using a compass, measure the length of the diagonal AC.
With the compass point at A, draw an arc on the perpendicular line from A. Label the intersection point P. So, AP has the same length as AC.
With the compass point at C, draw an arc on the perpendicular line from C. Label the intersection point Q. So, CQ has the same length as AC.
Connect points P and Q with a straight line.
The figure APCQ is the new square.
Its side length is AC, which is $s\sqrt{2}$ .
Its area is $(s\sqrt{2})^2 = 2s^2$ , which is double the area of the original square.

Diagram 3

Answer

To construct a square having double the area of a given square:

(i) Draw the given square and find the length of its diagonal. (ii) Use this diagonal as the side length for the new square. (iii) Construct a square with this diagonal length as its side.

Question 1

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