Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are of equal length.

Properties of an Equilateral Triangle

• All three sides are congruent
• All three angles are congruent and each measures 60 degrees
• The altitude, median, and angle bisector from any vertex are the same line
• The centroid, circumcenter, and orthocenter coincide at the center of the triangle
• The area can be calculated using the formula: Area = (s^2 * √3) / 4, where s is the length of a side
• The perimeter can be calculated using the formula: Perimeter = 3s, where s is the length of a side

Example Problems

1. Find the area and perimeter of an equilateral triangle with side length 8 cm.

Area = (8^2 * √3) / 4 = (64 * √3) / 4 = 16√3 cm²

Perimeter = 3 * 8 = 24 cm

2. If the height of an equilateral triangle is 10 cm, find the length of each side.

Using the formula for the area: Area = (s^2 * √3) / 4

10 = (s^2 * √3) / 4

s^2 = (10 * 4) / √3 = 40 / √3

s = √(40 / √3) = √(40 * √3) / 3 = 2√10 / √3 = (2√10 * √3) / 3√3 = (2√30) / 3

Conclusion

Equilateral triangles have special properties that make them unique. Understanding these properties and formulas can help in solving problems related to equilateral triangles.