If side of a square is equal to height of equilateral triangle, then find ratio of area of equilateral triangle to area of square?

Options

A. A)`sqrt2 : 5`
B. B)`3:5`
C. C)`sqrt3 :2`
D. D)`1 : sqrt3`

Answer

To solve the problem, we need to find the ratio of the area of an equilateral triangle to the area of a square, given that the side of the square is equal to the height of the equilateral triangle.

### Step-by-Step Solution:

1. **Define Variables:**
Let the side of the square be \( s \). According to the problem, the height of the equilateral triangle is also \( s \).

2. **Area of the Square:**
The area \( A_s \) of the square can be calculated using the formula:
\[ A_s = s^2 \]

3. **Height of the Equilateral Triangle:**
The height \( h \) of an equilateral triangle with side length \( a \) can be calculated using the formula:
\[ h = \frac{\sqrt{3}}{2} a \]
Since we know that the height \( h \) is equal to the side of the square \( s \), we can set:
\[ s = \frac{\sqrt{3}}{2} a \]
Rearranging gives:
\[ a = \frac{2s}{\sqrt{3}} \]

4. **Area of the Equilateral Triangle:**
The area \( A_t \) of the equilateral triangle can be calculated using the formula:
\[ A_t = \frac{\sqrt{3}}{4} a^2 \]
Substituting \( a = \frac{2s}{\sqrt{3}} \):
\[ A_t = \frac{\sqrt{3}}{4} \left(\frac{2s}{\sqrt{3}}\right)^2 \]
Simplifying this gives:
\[ A_t = \frac{\sqrt{3}}{4} \cdot \frac{4s^2}{3} = \frac{\sqrt{3}}{3} s^2 \]

5. **Ratio of Areas:**
Now, we need to find the ratio of the area of the triangle to the area of the square:
\[ \text{Ratio} = \frac{A_t}{A_s} = \frac{\frac{\sqrt{3}}{3} s^2}{s^2} \]
This simplifies to:
\[ \text{Ratio} = \frac{\sqrt{3}}{3} \]

6. **Final Ratio:**
To express this in a more standard form, we can multiply both the numerator and the denominator by 3:
\[ \text{Ratio} = 1 : \sqrt{3} \]

### Conclusion:
The ratio of the area of the equilateral triangle to the area of the square is \( 1 : \sqrt{3} \).