1. What is the Median of an Equilateral Triangle?

The median of an equilateral triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. In an equilateral triangle, all three medians are equal in length and coincide with the altitudes and angle bisectors.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( M \) — Median of Equilateral Triangle (m)
• \( h \) — Height of Equilateral Triangle (m)

Explanation: In an equilateral triangle, the median is equal to the height of the triangle. This relationship holds true because all sides and angles are equal in an equilateral triangle.

3. Properties of Medians in Equilateral Triangles

Details: In an equilateral triangle, the medians have several important properties: they are all equal in length, they intersect at a single point called the centroid, and this point divides each median in a 2:1 ratio. Additionally, the medians are also altitudes and angle bisectors in equilateral triangles.

4. Using the Calculator

Tips: Enter the height of the equilateral triangle in meters. The height must be a positive value greater than zero. The calculator will compute the median length, which will be equal to the input height.

5. Frequently Asked Questions (FAQ)

Q1: Why is the median equal to the height in an equilateral triangle?
A: In an equilateral triangle, all sides and angles are equal. The median, altitude, and angle bisector from any vertex coincide and have the same length due to the symmetry of the triangle.

Q2: Does this relationship hold for all types of triangles?
A: No, this specific relationship (median = height) is unique to equilateral triangles. In other types of triangles (isosceles, scalene), medians and heights are generally different.

Q3: How is the height related to the side length in an equilateral triangle?
A: The height (h) of an equilateral triangle with side length (a) can be calculated using the formula: \( h = \frac{\sqrt{3}}{2} \times a \).

Q4: Can I calculate the median if I know the side length instead of the height?
A: Yes, the median length can also be calculated directly from the side length using the formula: \( M = \frac{\sqrt{3}}{2} \times a \), where a is the side length.

Q5: Are there any practical applications of this calculation?
A: Yes, calculating medians in equilateral triangles is important in various fields including architecture, engineering, computer graphics, and any application involving geometric calculations with equilateral triangles.