Key Concepts and Formulas for Equilateral Triangles: Perimeter, Area, Height, and Relationships between Sides and Angles

equilateral triangle

An equilateral triangle is a type of triangle in which all three sides are equal in length

An equilateral triangle is a type of triangle in which all three sides are equal in length. In addition, all three angles are also equal and measure 60 degrees each. It is a special case of an isosceles triangle, where two sides are equal in length.

To solve problems related to equilateral triangles, here are some key concepts and formulas:

1) Perimeter: The perimeter of an equilateral triangle can be found by multiplying the length of one side by 3, since all three sides are equal. So, if the length of one side is “s”, the perimeter would be P = 3s.

2) Area: The area of an equilateral triangle can be found using the formula A = (√3/4) * s^2, where “s” is the length of one side. This formula derives from the fact that an equilateral triangle can be divided into two congruent right triangles.

3) Height: The height of an equilateral triangle is the perpendicular distance from any side to the opposite vertex. To calculate the height, you can use the formula h = (sqrt(3) / 2) * s, where “s” is the length of one side.

4) Relationships between sides and angles: Since all sides of an equilateral triangle are equal, the relationship between sides and angles can be determined using trigonometric functions. For example, the length of one side can be found by dividing the perimeter by 3. Then, you can use trigonometric functions like sin, cos, or tan to find the values of angles.

To better understand these concepts, let’s see an example problem:

Example: Find the perimeter, area, and height of an equilateral triangle if one side measures 6 cm.

1) Perimeter: P = 3s = 3 * 6 cm = 18 cm
The perimeter of the equilateral triangle is 18 cm.

2) Area: A = (√3/4) * s^2 = (√3/4) * 6^2 cm^2
A = (√3/4) * 36 cm^2
The area of the equilateral triangle is (√3/4) * 36 cm^2 or approximately 93.53 cm^2.

3) Height: h = (sqrt(3) / 2) * s = (sqrt(3) / 2) * 6 cm
The height of the equilateral triangle is (sqrt(3) / 2) * 6 cm or approximately 5.2 cm.

Remember to always double-check your calculations and units to ensure accuracy.

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