Equilateral Triangle Perpendicular Length Calculator
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An equilateral triangle is a triangle in which all three sides are equal in length, and all three angles are also equal to 60 degrees. It is a special case of the more general class of triangles known as regular polygons. One of the key properties of an equilateral triangle is that its perpendicular height, or altitude, is equal to the length of one of its sides. In this post, we will explore the concept of the perpendicular height of an equilateral triangle in more detail.
Perpendicular Height of an Equilateral Triangle
The perpendicular height of an equilateral triangle is a line segment drawn from one of its vertices to the opposite side, perpendicular to that side. This line segment is also known as the altitude of the triangle. The altitude divides the equilateral triangle into two congruent 30-60-90 right triangles, as shown in the diagram below:
In the diagram, the altitude is labeled as h, and the side length of the equilateral triangle is labeled as s. The perpendicular height of an equilateral triangle can be calculated using the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side (the hypotenuse).
Using the Pythagorean theorem, we can derive the formula for the perpendicular height of an equilateral triangle:
h = (sqrt(3) / 2) * s
where h is the perpendicular height and s is the length of a side of the equilateral triangle.
Perimeter of an Equilateral Triangle
The perimeter of an equilateral triangle is simply the sum of the lengths of its three sides. Since all three sides of an equilateral triangle are equal in length, we can simply multiply the length of one side by three to find the perimeter. Therefore, the formula for the perimeter of an equilateral triangle is:
P = 3s
where P is the perimeter and s is the length of a side of the equilateral triangle.
Area of an Equilateral Triangle
The area of an equilateral triangle can be calculated using the formula:
A = (sqrt(3) / 4) * s^2
where A is the area and s is the length of a side of the equilateral triangle.
The formula for the area of an equilateral triangle can be derived by dividing the triangle into two 30-60-90 right triangles, as shown in the diagram below:
In the diagram, the altitude is labeled as h, and the side length of the equilateral triangle is labeled as s. The area of each 30-60-90 right triangle can be calculated using the formula for the area of a triangle:
A = (1 / 2) * base * height
where base is the length of the side of the equilateral triangle and height is the perpendicular height (altitude) of the equilateral triangle. Therefore, the total area of the equilateral triangle is the sum of the areas of the two 30-60-90 right triangles:
A = 2 * (1 / 2) * s * h A = s * h
Substituting the formula for the perpendicular height of an equilateral triangle, we get:
A = (sqrt(3) / 2) * s * (s / 2) A = (sqrt(3) / 4) * s^2
Here are some examples with solutions related to the perpendicular length of an equilateral triangle:
Example 1: Find the perpendicular length of an equilateral triangle with side length 6 cm.
Solution: Let’s draw an equilateral triangle ABC with side length 6 cm, where the perpendicular line from vertex A intersects the opposite side at point D. We know that AD is the perpendicular height of the equilateral triangle.
Now, we can draw a line from vertex A to the midpoint E of BC, which will bisect the base BC into two equal parts. We can use this line AE to form a right triangle AED, where DE is the perpendicular height and AD is the hypotenuse.
Since we know that the side length of the equilateral triangle is 6 cm, we can use the Pythagorean theorem to find the value of DE:
DE^2 + AE^2 = AD^2 DE^2 + (BC/2)^2 = 6^2 DE^2 + (3)^2 = 36 DE^2 = 36 – 9 DE^2 = 27 DE = sqrt(27) DE = 3sqrt(3) cm
Therefore, the perpendicular height of the equilateral triangle is 3sqrt(3) cm.
Example 2: The area of an equilateral triangle is 48 square meters. Find the length of the perpendicular height.
Solution: Let’s denote the side length of the equilateral triangle as s, and the perpendicular height as h. We know that the area of an equilateral triangle is given by the formula:
Area = (sqrt(3)/4) * s^2
We are given that the area is 48 square meters, so we can set up an equation:
48 = (sqrt(3)/4) * s^2
Multiplying both sides by 4/sqrt(3), we get:
64/sqrt(3) = s^2
Taking the square root of both sides, we get:
s = sqrt(64/sqrt(3)) s = 8/sqrt(3)
Now, we can use the side length to find the perpendicular height h. We can draw an equilateral triangle with side length 8/sqrt(3) and use the same method as in Example 1 to find the value of h:
h = (sqrt(3)/2) * s h = (sqrt(3)/2) * (8/sqrt(3)) h = 4 meters
Therefore, the length of the perpendicular height is 4 meters.
