If each side of triangle is increased by 15,% then by what % new area will increase in iqiulateral triangle

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.

If each side of a triangle is increased by 15%, then by what % will the new area increase in an equilateral triangle?

Discover how increasing each side of an equilateral triangle by 15% affects its area. This detailed guide walks through the mathematical concepts of scaling and area calculation, showing that the area increases by 32.25%. Perfect for high school geometry students.

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