The question involves a square with 4 equilateral triangles attached to its sides. The side of the square is given as 5 cm. We are tasked with finding the area of the entire figure, which includes both the square and the four equilateral triangles.

Step-by-step breakdown:

1. Area of the square: The side of the square is 5 cm. The area of a square is calculated using the formula: $A_{\text{square}} = s^2$ where $s$ is the side of the square. $A_{\text{square}} = 5^2 = 25 \, \text{cm}^2$

2. Area of one equilateral triangle: The side of each equilateral triangle is the same as the side of the square, which is 5 cm. The area $A$ of an equilateral triangle with side $a$ is given by the formula: $A_{\text{triangle}} = \frac{\sqrt{3}}{4} a^2$ Substituting $a = 5$: $A_{\text{triangle}} = \frac{\sqrt{3}}{4} \times 5^2 = \frac{\sqrt{3}}{4} \times 25 = 6.25 \sqrt{3} \, \text{cm}^2$

   Using $\sqrt{3} \approx 1.732$: $A_{\text{triangle}} = 6.25 \times 1.732 = 10.825 \, \text{cm}^2$

3. Total area of four equilateral triangles: Since there are four equilateral triangles: $A_{\text{4 triangles}} = 4 \times 10.825 = 43.3 \, \text{cm}^2$

4. Total area of the figure: The total area of the figure is the area of the square plus the area of the four equilateral triangles: $A_{\text{total}} = A_{\text{square}} + A_{\text{4 triangles}} = 25 + 43.3 = 68.3 \, \text{cm}^2$

Final Answer:

The total area of the figure is approximately 68.3 cm².

Would you like further details on any of the steps?

Related Questions:

1. How do you calculate the height of an equilateral triangle given its side?
2. What is the perimeter of the entire figure?
3. How would the area change if the side length of the square were doubled?
4. What are the properties of an equilateral triangle that are useful in geometry problems?
5. How can you verify the result using geometry software or graphing tools?

Tip:

When solving geometry problems, sketching the figure and labeling all known values can help visualize the relationships between the shapes.