Try to think of different creative ways to divide a square into 4 parts of equal area.

Question

Homi · Accepted Answer

We will explore different geometric constructions to divide a square into four parts, each having one-fourth of the total area.

Step 1 — Four Smaller Squares

We can divide the square using lines parallel to its sides. Let the side length of the square be $s$ . The total area of the square is $A = s \times s = s^2$ . We draw a horizontal line connecting the midpoints of the vertical sides. We draw a vertical line connecting the midpoints of the horizontal sides. These lines divide the original square into four smaller squares. Each smaller square has a side length of $s/2$ . Let $A_1$ be the area of one small square.

$A_1 = \left(\frac{s}{2}\right) \times \left(\frac{s}{2}\right)$

$= \frac{s^2}{4}$

$\boxed{\text{Each part has area } \frac{s^2}{4}}$

Diagram 1

Step 2 — Four Triangles

We can divide the square by drawing its two diagonals. Let the side length of the square be $s$ . The total area of the square is $A = s^2$ . We draw the two diagonals of the square. The diagonals intersect at the center of the square. This divides the square into four congruent (identical in shape and size) triangles. Let us consider one such triangle. Its base is one side of the square. So its base length is $s$ . Its height is the perpendicular distance from the center to that side. This height is half the side length of the square. So it is $s/2$ . Let $A_2$ be the area of one triangle.

$A_2 = \frac{1}{2} \times \text{base} \times \text{height}$

$= \frac{1}{2} \times s \times \frac{s}{2}$

$= \frac{s^2}{4}$

$\boxed{\text{Each part has area } \frac{s^2}{4}}$

Diagram 2

Step 3 — Four Quadrilaterals (Pinwheel Pattern)

We can connect the square's center to the midpoints of its sides. Let the side length of the square be $s$ . The total area of the square is $A = s^2$ . Let O be the center of the square. Let M1, M2, M3, M4 be the midpoints of sides AB, BC, CD, DA. We draw lines from center O to each midpoint. This divides the square into four congruent quadrilaterals (four-sided shapes). Let us consider one such quadrilateral, for example, AM1OM4. This quadrilateral splits into two triangles: $\triangle$ AOM1 and $\triangle$ AOM4. For $\triangle$ AOM1, its base AM1 has length $s/2$ . Its height is the perpendicular distance from O to AB. This height is $s/2$ . Area of $\triangle$ AOM1 is $A_{\triangle \text{AOM1}} = \frac{1}{2} \times \frac{s}{2} \times \frac{s}{2} = \frac{s^2}{8}$ . For $\triangle$ AOM4, its base AM4 has length $s/2$ . Its height is the perpendicular distance from O to AD. This height is $s/2$ . Area of $\triangle$ AOM4 is $A_{\triangle \text{AOM4}} = \frac{1}{2} \times \frac{s}{2} \times \frac{s}{2} = \frac{s^2}{8}$ . Let $A_3$ be the area of the quadrilateral AM1OM4.

$A_3 = A_{\triangle \text{AOM1}} + A_{\triangle \text{AOM4}}$

$= \frac{s^2}{8} + \frac{s^2}{8}$

$= \frac{2s^2}{8}$

$= \frac{s^2}{4}$

$\boxed{\text{Each part has area } \frac{s^2}{4}}$

Diagram 3

Answer

The three creative ways to divide a square into 4 parts of equal area are:

(i) Dividing it into four smaller congruent squares by connecting the midpoints of opposite sides. (ii) Dividing it into four congruent triangles by drawing its two diagonals. (iii) Dividing it into four congruent quadrilaterals by connecting the center of the square to the midpoints of its sides.

Question 1

Answer

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