To find the altitude of an equilateral triangle with each side measuring 'a' cm, we can use geometric relations.

Understanding the Equilateral Triangle
- An equilateral triangle has all three sides of equal length: \( a \).
- All internal angles are \( 60^\circ \).

Visualizing the Altitude
- The altitude (height) of an equilateral triangle can be drawn from one vertex to the midpoint of the opposite side.
- This altitude divides the triangle into two 30-60-90 right triangles.

Using the Pythagorean Theorem
- In a 30-60-90 triangle, the sides are in the ratio \( 1 : \sqrt{3} : 2 \).
- The side opposite the \( 30^\circ \) angle is \( \frac{a}{2} \) (half of the base).
- The side opposite the \( 60^\circ \) angle is the altitude (h).

Calculating the Altitude
1. **Identify the sides**:
- Base = \( \frac{a}{2} \)
- Hypotenuse = \( a \)
- Altitude = \( h \)
2. **Apply the Pythagorean theorem**:
- \( a^2 = h^2 + \left(\frac{a}{2}\right)^2 \)
3. **Simplifying**:
- \( a^2 = h^2 + \frac{a^2}{4} \)
- Rearranging gives \( h^2 = a^2 - \frac{a^2}{4} = \frac{3a^2}{4} \)
4. **Finding h**:
- \( h = \sqrt{\frac{3a^2}{4}} = \frac{a\sqrt{3}}{2} \)

Final Result
- The altitude \( h \) of an equilateral triangle with side \( a \) is given by:
**\[ h = \frac{a\sqrt{3}}{2} \]**