Understanding the Height of an Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are of equal length. One of the important properties of an equilateral triangle is its height, which can be calculated using various methods. In this article, we will explore how to find the height of an equilateral triangle with a given side length, specifically using the case where the side length is 2 cm.

Calculating the Height of an Equilateral Triangle

The height h of an equilateral triangle can be calculated using the formula:

h (sqrt{3}/2) * a

where a is the length of a side of the triangle. For a side length of 2 cm:

h (sqrt{3}/2) * 2 sqrt{3} cm ≈ 1.73 cm

Therefore, the height of the equilateral triangle with a side length of 2 cm is approximately 1.73 cm.

Altitude and Geometrical Facts

The three sides of an equilateral triangle are equal, and each side is denoted as a. When side length is known, one can find its area and altitude. For instance, if a 2 cm:

The altitude (or height) can be calculated as: Altitude 2 sin(60°) (2 * sqrt{3})/2 sqrt{3} The area of the triangle can be found using the formula: Area (a * h) / 2 (2 * sqrt{3})/2 sqrt{3} cm2

Derivations and Proving the Height

Using the properties of a 30°-60°-90° triangle:

Applying the Pythagorean Theorem: Height h sqrt{a^2 - (a/2)^2} sqrt{3}a/2 Therefore, the height of an equilateral triangle with side length a is: h (sqrt{3}/2) * a (sqrt{3}/2) * 2 sqrt{3} cm ≈ 1.73 cm

Practical Exercises

Here are a couple of exercises to help you practice finding the height of an equilateral triangle with different side lengths:

What is the height of an equilateral triangle of side 10 cm? What is the height of an equilateral triangle of side 15 cm?

To find the height, use the derived formula or methods such as the Pythagorean Theorem or trigonometric functions.

Method 1 - Using the Pythagorean Theorem

In an equilateral triangle, the height also acts as the median, splitting the base into two equal segments. Thus, if we know one side length (2 cm), we can find the height:

Height sqrt{2^2 - 1^2} sqrt{3} cm ≈ 1.73 cm

Thus, the height is again approximately 1.73 cm.

Method 2 - Using Heron's Formula

Heron's formula for the area of a triangle can be used:

A sqrt{s(s-a)(s-b)(s-c)}

where s is the semi-perimeter and a, b, c are the sides of the triangle. For an equilateral triangle with side length of 2 cm:

Semi-perimeter (2 2 2)/2 3 cm A sqrt{3(3-2)(3-2)(3-2)} sqrt{3} cm2 Height (2 * A) / a (2 * sqrt{3}) / 2 sqrt{3} cm ≈ 1.73 cm

This confirms the height is again approximately 1.73 cm.