{"id":3301,"date":"2026-04-13T05:46:23","date_gmt":"2026-04-13T05:46:23","guid":{"rendered":"https:\/\/www.mathros.net.ua\/en\/?p=3301"},"modified":"2026-07-18T13:05:57","modified_gmt":"2026-07-18T13:05:57","slug":"height-of-an-equilateral-triangle","status":"publish","type":"post","link":"https:\/\/www.mathros.net.ua\/en\/height-of-an-equilateral-triangle.html","title":{"rendered":"Height of an Equilateral Triangle: Formula and Examples"},"content":{"rendered":"

Height of an equilateral triangle is a segment that connects a vertex of the triangle to the opposite side. In an equilateral triangle, the height has a special role. It is not only perpendicular to the side, but it also divides that side into two equal parts. Because of this, the height splits the equilateral triangle into two equal right triangles, and the formula for finding it can be conveniently derived using the Pythagorean theorem.<\/p>\n

In this article, we will look at how to find the height of an equilateral triangle, how the corresponding formula is derived, and how to apply it when solving practical problems.<\/p>\n

Height of an Equilateral Triangle: Main Formula<\/h2>\n

An equilateral triangle is a triangle in which all sides have the same length. If the side length is known, then the height of such a triangle can be found using the formula:<\/p>\n

\\[
\nBH=\\frac{\\sqrt{3}\\cdot AB}{2}.
\n\\]<\/p>\n

So, to find the height of an equilateral triangle, it is enough to know the length of just one of its sides.<\/p>\n

\"Image:<\/p>\n

Proof of the Formula<\/h3>\n

As already mentioned, the formula for the height of an equilateral triangle can be obtained using the Pythagorean theorem.<\/p>\n

Recall that according to the Pythagorean theorem, the sum of the squares of the legs is equal to the square of the hypotenuse. Consider triangle \\( ABC \\). Since it is equilateral, all its sides are equal. In addition, the height \\( BH \\), drawn to side \\( AC \\), is also a median in such a triangle. This means that it divides side \\( AC \\) into two equal parts. Therefore,<\/p>\n

\\[
\nAH=\\frac{AB}{2}.
\n\\]<\/p>\n

Now consider one of the two right triangles formed, for example \\( ABH \\). For this triangle, by the Pythagorean theorem, we have:<\/p>\n

\\[
\nAB^2=AH^2+BH^2.
\n\\]<\/p>\n

Substitute \\( AH=\\frac{AB}{2} \\):<\/p>\n

\\[
\nAB^2=\\left(\\frac{AB}{2}\\right)^2+BH^2=\\frac{AB^2}{4}+BH^2.
\n\\]<\/p>\n

Now express \\( BH \\) from this equation:<\/p>\n

\\[
\nBH^2=AB^2-\\frac{AB^2}{4},\\qquad BH^2=\\frac{3\\cdot AB^2}{4},\\qquad BH=\\frac{\\sqrt{3}\\cdot AB}{2}.
\n\\]<\/p>\n

Thus, the formula for finding the height of an equilateral triangle is proved.<\/p>\n

Note<\/strong>. If we denote the side length of the triangle by \\( a \\), and the height by \\( h \\), then the formula takes the more familiar form:
\n\\[
\nh=\\frac{\\sqrt{3}\\cdot a}{2}.
\n\\]<\/p><\/blockquote>\n

Height of an Equilateral Triangle: Examples with Answers<\/h2>\n

The examples below are solved using the formula for the height of an equilateral triangle. Try to do the calculations on your own first, and then compare them with the ready-made solutions.<\/p>\n

Example 1. Find the height of an equilateral triangle whose side is \\( 7 \\) cm<\/h3>\n

We use the height formula with \\( a=7 \\):<\/p>\n

\\[
\nh=\\frac{\\sqrt{3}\\cdot a}{2}=\\frac{\\sqrt{3}\\cdot 7}{2}\\approx 6.062.
\n\\]<\/p>\n

So, the height of an equilateral triangle with side \\( 7 \\) cm is \\( 6.062 \\) cm.<\/p>\n

Example 2. What is the height of an equilateral triangle with side \\( 9 \\) cm?<\/h3>\n

According to the problem, we have \\( a=9 \\). Substitute this value into the formula:<\/p>\n

\\[
\nh=\\frac{\\sqrt{3}\\cdot a}{2}=\\frac{\\sqrt{3}\\cdot 9}{2}\\approx 7.794.
\n\\]<\/p>\n

Therefore, the height of the equilateral triangle is \\( 7.794 \\) cm.<\/p>\n

Example 3. If the height of an equilateral triangle is \\( 6 \\) cm, what is the length of one of its sides?<\/h3>\n

In this case, the height is known, and we need to find the side. We take the formula<\/p>\n

\\[
\nh=\\frac{\\sqrt{3}\\cdot a}{2}
\n\\]<\/p>\n

and express \\( a \\) from it:<\/p>\n

\\[
\nh=\\frac{\\sqrt{3}\\cdot a}{2},\\qquad 6=\\frac{\\sqrt{3}\\cdot a}{2},\\qquad 12=\\sqrt{3}\\cdot a,\\qquad a\\approx 6.928.
\n\\]<\/p>\n

So, the length of one side of the triangle is \\( 6.928 \\) cm.<\/p>\n

Example 4. What is the height of an equilateral triangle if its perimeter is \\( 63 \\) cm?<\/h3>\n

The perimeter is the sum of the lengths of the three sides of a triangle. Since all sides in an equilateral triangle are equal, we first find the length of one side:<\/p>\n

\\[
\na=\\frac{63}{3}=21.
\n\\]<\/p>\n

Now substitute the found value into the height formula:<\/p>\n

\\[
\nh=\\frac{\\sqrt{3}\\cdot a}{2}=\\frac{\\sqrt{3}\\cdot 21}{2}\\approx 18.186.
\n\\]<\/p>\n

Thus, the height of the equilateral triangle is \\( 18.186 \\) cm.<\/p>\n

Example 5. Find the height of an equilateral triangle if its area is \\( 60\\ \\text{cm}^2 \\)<\/h3>\n

Here, we cannot substitute the value directly into the height formula because the side of the triangle is still unknown. So first, we find the side using the area formula for an equilateral triangle:<\/p>\n

\\[
\nA=\\frac{\\sqrt{3}\\cdot a^2}{4}.
\n\\]<\/p>\n

Substitute the given area value:<\/p>\n

\\[
\n60=\\frac{\\sqrt{3}\\cdot a^2}{4},\\qquad 240=\\sqrt{3}\\cdot a^2,\\qquad a\\approx 11.771.
\n\\]<\/p>\n

Now, when the side has already been found, we use the height formula:<\/p>\n

\\[
\nh=\\frac{\\sqrt{3}\\cdot a}{2}=\\frac{\\sqrt{3}\\cdot 11.771}{2}\\approx 10.194.
\n\\]<\/p>\n

So, the height of the equilateral triangle is \\( 10.194 \\) cm.<\/p>\n

What to Read Next: Useful Topics to Continue With<\/h2>\n

Would you like to understand the topic of the equilateral triangle even better? Then it is worth exploring related topics. They will help you see this figure<\/a> more broadly and better understand its main properties.<\/p>\n

    \n
  1. Equilateral Triangle: Definition and Properties<\/a> \u2014 Learn what the main properties of an equilateral triangle are and how it differs from other triangles.<\/li>\n
  2. Perimeter of an Equilateral Triangle: Formulas and Examples<\/a> \u2014 This topic shows how to calculate the perimeter of an equilateral triangle when the side length is known.<\/li>\n
  3. Area of an Equilateral Triangle: Formulas and Examples<\/a> \u2014 Here you will see how to find the area of an equilateral triangle and how to apply the corresponding formula in practice.<\/li>\n<\/ol>\n

    Height and Code: Try to Implement the Algorithm Yourself<\/h2>\n

    If you are interested not only in solving geometry problems but also in programming, take a look at the flowchart below. It clearly shows how a mathematical formula turns into a simple and understandable algorithm that can be easily implemented in any programming language you are comfortable with. Why not use this chart as the basis for your own small program? This kind of practice helps you better understand the formula for the height of an equilateral triangle, see how it is used in calculations, and at the same time take one more step in learning programming.<\/p>\n

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