Height of Equilateral Triangle
The height of an equilateral triangle is a straight line that is drawn from the vertex to the opposite side of the triangle in such a way that it divides the triangle into two equal right-angled triangles. This is also known as the altitude of the triangle which starts from the vertex and is the perpendicular bisector of the opposite side. An equilateral triangle is a triangle in which all the sides are of equal length and all the angles are of equal measure. Let us learn more about the height of equilateral triangle in this article.
| 1. | What is Height of Equilateral Triangle? |
| 2. | Height of Equilateral Triangle Formula |
| 3. | FAQs on Height of Equilateral Triangle |
What is Height of Equilateral Triangle?
The height of an equilateral triangle is a line that is drawn from any vertex of the triangle on the opposite side. This line is the perpendicular bisector of the opposite side. This means it bisects the opposite side into two equal parts and forms an angle of 90° on it. The height of an equilateral triangle is also known as the altitude which divides the triangle into two congruent right-angled triangles as shown in the following figure.
Equilateral Triangle Definition
An Equilateral triangle is defined as a triangle where all three sides and angles are equal. The value of each angle is 60 degrees, therefore, it is also known as an equiangular triangle. The equilateral triangle is considered as a regular polygon because all its angles and sides are equal.
Height of Equilateral Triangle Formula
The height of equilateral triangle can be calculated with the help of the Pythagoras theorem. The formula that is used to find the height of an equilateral triangle is,
Height of equilateral triangle (h) = ½(√3a) or (a√3)/2
where, 'a' is the side of the equilateral triangle. This means if the side length of the equilateral triangle is known, we can easily find the height of the triangle.
Example: If one side of an equilateral triangle is 12 units, what is its height?
Solution: Given, side length, a = 12 units.
We will use the formula for the height of equilateral triangle, h = (a√3)/2. Substituting the value of 'a', we get, h = (12√3)/2 = 6√3 = 10.39 units. Therefore, the height of the equilateral triangle is 10.39 units.
Height of Equilateral Triangle Formula Proof
We know that the altitude splits the equilateral triangle into two right-angled triangles. Therefore, we will apply the Pythagoras theorem, which says Hypotenuse2 = Base2 + Height2
If we observe the figure given above, each side of the equilateral triangle is represented by the letter 'a', since all the sides are equal, so, the base is also represented by the letter 'a', and height = h. It is also known that the altitude divides the triangle into two congruent right-angled triangles and the base is divided into two equal parts. This means that if we use the Pythagoras theorem in one of the right-angled triangles, this can be expressed as,
a2 = h2 + (a/2)2
⇒ h2 = a2 - (a2/4)
⇒ h2 = (3a2)/4
Or, h = ½(√3a)
Therefore, the formula for the height of an equilateral triangle is,
Height of Equilateral Triangle if Side is Given
The height of an equilateral triangle can be easily calculated if the side is given. Since all the sides of an equilateral triangle are equal and we know that the altitude divides the triangle into congruent right-angled triangles, we can apply the Pythagoras theorem and find the height as explained in the section given above.
Therefore, if one side of the equilateral triangle is given, the formula to find the height of equilateral triangle is, h = ½(√3a), where 'a' represents the side length of the equilateral triangle.
Height of Equilateral Triangle when Area is Given
The height of an equilateral triangle can be calculated when the area of the triangle is given. We know that the area of an equilateral triangle can be calculated with the formula, Area of equilateral triangle = √3/4 × (side)2. So, if we know the area, we can substitute its value in this formula to get the side length. Once the side length is known, we can use the following formula to find the height. Height of equilateral triangle, h = ½(√3a).
Example: Find the height of an equilateral triangle if its area is 24 square units.
Solution: Given, area of the equilateral triangle = 24 unit2
First, we will find the side length using the formula, Area of equilateral triangle = √3/4 × (side)2
24 = (√3)/4 × (side)2
(side)2 = (24 × 4)/√3
side = 7.4 units
Now, after we know the side length, we can calculate the height of the equilateral triangle using the formula, h = ½(√3a), where 'h' is the height and 'a' represents the side length. Here, 'a' = 7.4 units
h = ½(√3a)
h = ½(√3 × 7.4) = 6.4 units.
Height of Equilateral Triangle when Perimeter is Given
The height of an equilateral can be calculated if the perimeter of the triangle is given. We know that the perimeter of an equilateral triangle is calculated by the formula, perimeter of equilateral triangle = 3a. It is also known that all the sides of an equilateral triangle are equal in length, therefore, if the perimeter is known, we can calculate the side length using this formula. After the side length is calculated we can find the height using the formula, height of equilateral triangle = ½(√3a)
Example: Find the height of an equilateral triangle if its perimeter is 21 units.
Solution: Given, the perimeter of the equilateral triangle = 21 units
First, we will find the side length using the formula, Perimeter of equilateral triangle = 3a
21 = 3a
a = 7 units
Now, we can calculate the height of the equilateral triangle using the formula, h = ½(√3a), where 'h' is the height and 'a' represents the side length. We have calculated the side length as, 'a' = 7 units
h = ½(√3a)
h = ½(√3 × 7) = 6.06 units
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FAQs on Height of Equilateral Triangle
What is the Height of Equilateral Triangle in Math?
The height of an equilateral triangle is a straight line that is drawn from the vertex to the opposite side of the triangle in such a way that it divides the triangle into two equal right-angled triangles. This is also known as the altitude of the triangle which starts from the vertex and is the perpendicular bisector of the opposite side.
What is the Formula of Height of Equilateral Triangle?
The height of an equilateral triangle can be calculated using the Pythagoras theorem. The formula that is used to find the height of an equilateral triangle is, Height of equilateral triangle (h) = (a√3)/2; where 'a' is the side of the equilateral triangle. This means if the side length of the equilateral triangle is known, we can easily find the height of the triangle.
How to Find the Height of Equilateral Triangle if Side is Given?
The height of an equilateral can be easily calculated if the side is given. We know that all the sides of an equilateral triangle are equal and the altitude divides the triangle into two congruent right-angled triangles. Therefore, we can apply the Pythagoras theorem and find the height. The formula to find the height of an equilateral triangle is, h = (a√3)/2, where 'a' represents the side length of the equilateral triangle.
How to Find the Height of Equilateral Triangle When Area is Given?
The height of an equilateral triangle can be calculated when the area of the triangle is given. We know that the area of an equilateral triangle can be calculated with the formula, Area of equilateral triangle = √3/4 × (side)2. So, if we know the area, we can substitute its value in this formula to get the side length. Once the side length is known, we can use the following formula to find the height. Height of equilateral triangle, h = ½(√3a).
How to Find the Height of Equilateral Triangle When Perimeter is Given?
When the perimeter of an equilateral is given, we can find the height easily. The formula that is used to find the perimeter of an equilateral triangle is, perimeter = 3a, where 'a' represents the side length. Using this formula, we can find the side length after substituting the value of the perimeter. After the side length is calculated, we can use the formula, Height of equilateral triangle, h = ½(√3a), to find the height of the equilateral triangle.
How to Find the Height of an Equilateral Triangle Using Pythagorean Theorem?
We know that the altitude splits the equilateral triangle into two right-angled triangles. Therefore, we can use the Pythagoras theorem to find the height of an equilateral triangle. The Pythagoras theorem says Hypotenuse2 = Base2 + Height2
Therefore, let us take one of the right-angled triangles, in which the side length (hypotenuse, in this case) = a, h = height of the triangle, and a/2 is the third side. Using the Pythagoras theorem, the formula for the height of the equilateral can be derived and expressed as follows,
a2 = h2 + (a/2)2
⇒ h2 = a2 – (a2/4)
⇒ h2 = (3a2)/4
Or, h = ½(√3a)
What is the Height of an Equilateral Triangle with Side Length 6 units?
If the side length of an equilateral is given as 6 units, its height can be calculated with the formula, Height of equilateral triangle, h = ½(√3a), where 'a' represents the side length. Substituting the value of 'a' in the formula,
Height of equilateral triangle, h = ½(√3a) = ½(√3 × 6) = 3√3 = 5.19 units.
How to Find the Area of Equilateral Triangle with Height?
The area of an equilateral triangle can be calculated if the height of the triangle is given. We use the formula, h = ½(√3a), where 'h' is the height and 'a' represents the side length. Substituting the value of the given height 'h' in the formula, we can find the side 'a' of the triangle, and then the area can be calculated. Let us understand this with an example.
- For example, if the height of an equilateral triangle is 8 units, let us find the side of the triangle using the formula, h = ½(√3a)
- After substituting the value of h = 8, we get, h = ½(√3a) and the value of a = 9.32 units.
- Now, the area of the triangle can be calculated using the formula, Area of equilateral triangle = √3/4 × (side)2
- After substituting the values, we get Area of equilateral triangle = √3/4 × (9.32)2 = 37.61 square units.
How to Find the Altitude of an Equilateral Triangle?
The altitude of a triangle is another name for the height of a triangle. Therefore, the altitude of an equilateral triangle can be calculated using the formula, h = ½(√3a), where 'h' is the height and 'a' represents the side length. This means if the side length of an equilateral triangle is known, then the altitude can be easily calculated using this formula.
What is the Height of a Triangle?
The height of a triangle is the perpendicular line segment drawn from the vertex of the triangle to the opposite side. This height forms a right angle with the base of the triangle that it touches. It is denoted by the letter 'h' and is also known as the altitude of a triangle. It is to be noted that since there are three sides in a triangle, 3 altitudes can be drawn in it. Different triangles have different kinds of altitudes.