What is the smallest perimeter of a square if it can be divided without remainder into rectangles with sides 18 cm and 8 cm?

Let’s start by recalling the perimeter formula.

In geometry, the perimeter of any shape is equal to the sum of the lengths of its sides.

A rectangle has 4 sides: 2 lengths and 2 widths.

Let’s designate the length as a, width as b. The perimeter is indicated by a capital letter P.

From here we get:

P = a + a + b + b = 2 * a + 2 * b = 2 * (a + b);

So, the formula for the perimeter of a rectangle is:

P = 2 * (a + b);

Formula for the perimeter of a square
The formula for the perimeter of a square is found in a similar way. Since all its sides are equal, we take 1 side of it as a. Then the perimeter of the square is:

P = a + a + a + a = 4 * a;

If a square has all 4 sides equal, therefore, the length of its sides must be a multiple of the length and width of the rectangles that form this square.

In this case, the side of the square must be a multiple of the length (18 cm) and width (8 cm) of the rectangles into which it is divided. That is, the side of the square must be divided simultaneously by the length of the rectangles and the width of the rectangles without a remainder.

So, let’s find the least common multiple (LCM) of numbers 18 and 8.

LCM (18; 8) = 72;

So, the side of the square must be at least 72 cm.It can also be equal to:

72 * 2 = 144 cm;
72 * 3 = 216 cm;
72 * 4 = 288 cm, etc.
The smallest possible value is 72.

Let’s find the perimeter of this square with a side of 72 cm.It will be equal to:

P = 4 * a = 4 * 72 = 288 cm.

Answer: 288 cm.