An equilateral triangle with side 10.4 cm and a square have the same perimeter. Work out the side length of the square.

Step-by-step solution

1. 1

   Identify what "same perimeter" gives you. The two shapes have different side counts but a shared total boundary length, so the plan is: compute the triangle's perimeter first, then split that same total across the square's four sides.

2. 2

   Find the triangle's perimeter. An equilateral triangle has three equal sides, so $P=3\times 10.4=31.2$ cm.

3. 3

   Transfer the perimeter to the square. The square's perimeter is also $31.2$ cm.

4. 4

   Divide by the number of sides. A square has four equal sides, so $4s=31.2$ and $s=\dfrac{31.2}{4}=7.8$ cm.

5. 5

   Check the answer both ways. Multiplying back, $4\times 7.8=31.2$ cm, which matches $3\times 10.4=31.2$ cm. As a sense check, the square's side should be smaller than the triangle's because the same perimeter is shared among more sides, and indeed $7.8<10.4$ — exactly the ratio $\tfrac34$.