In an equilateral triangle, the area $A$ is related to the side length $s$ by the formula:

$A = \frac{\sqrt{3}}{4} s^2$

Step 1: Original Area

Let the original side length of the equilateral triangle be $s$. The original area $A_1$ is:

$A_1 = \frac{\sqrt{3}}{4} s^2$

Step 2: New Side Length

When each side is increased by 15%, the new side length $s'$ becomes:

$s' = s + 0.15s = 1.15s$

Step 3: New Area

The new area $A_2$ with the new side length $s'$ is:

$A_2 = \frac{\sqrt{3}}{4} (s')^2 = \frac{\sqrt{3}}{4} (1.15s)^2$ $A_2 = \frac{\sqrt{3}}{4} \times 1.3225s^2$ $A_2 = 1.3225 \times A_1$

Step 4: Percentage Increase in Area

The percentage increase in area is:

$\text{Percentage Increase} = \left( \frac{A_2 - A_1}{A_1} \right) \times 100 = \left( \frac{1.3225A_1 - A_1}{A_1} \right) \times 100$ $\text{Percentage Increase} = (1.3225 - 1) \times 100 = 0.3225 \times 100 = 32.25\%$

Conclusion:

The new area will increase by 32.25%.

Would you like further clarification on any of the steps?

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Here are 5 related questions you might find interesting:

1. What is the effect of increasing all sides of a right triangle by 15% on the area?
2. How does the area of an equilateral triangle relate to its height?
3. What is the relationship between the perimeter and area of an equilateral triangle?
4. How would decreasing the side length of an equilateral triangle affect the area?
5. Can the same formula be applied for an isosceles triangle?

Tip: When side lengths scale, areas scale by the square of the scale factor, hence small percentage increases in length result in larger percentage increases in area.