The Area of Triangles Formed by Joining the Diagonals of a Square

To find the area of the triangles formed by joining all of the diagonals of a square, we can follow a step-by-step process. Let's break it down:

Step 1: Understand the Problem
We are given a square with a side length of 4 cm. Our task is to find the area of the triangles formed by joining all of its diagonals.

Step 2: Identify the Diagonals
A square has two diagonals: one connecting opposite corners and another connecting the other two opposite corners. In total, there are four diagonals.

Step 3: Calculate the Length of the Diagonals
To find the length of the diagonals, we can use the Pythagorean theorem. Since the side length of the square is given as 4 cm, the length of the diagonals can be calculated as follows:

Diagonal = √(Side^2 + Side^2) = √(4^2 + 4^2) = √(16 + 16) = √32 cm

Step 4: Divide the Square into Four Triangles
By joining all of the diagonals, we can divide the square into four congruent triangles. Each triangle will have a base equal to the length of one side of the square (4 cm) and a height equal to half the length of one diagonal (0.5 * √32 cm).

Step 5: Calculate the Area of One Triangle
To find the area of one triangle, we can use the formula for the area of a triangle:

Area = (Base * Height) / 2 = (4 cm * 0.5 * √32 cm) / 2 = (2 cm * √32 cm) / 2 = √32 cm²

Step 6: Calculate the Total Area of All Four Triangles
Since there are four congruent triangles, the total area of all four triangles can be found by multiplying the area of one triangle by four:

Total Area = 4 * √32 cm² = 4√32 cm²

Therefore, the area of the triangles formed by joining all of the diagonals of the square is 4√32 cm².