Equilateral triangle calculator

How to use the equilateral triangle calculator?

1. Enter the side length of an equilateral triangle Area ({ \color{Orange} S }), Perimeter ({ \color{Orange} p }), Side Length ({ \color{Orange} h }) or Height ({ \color{Orange} a }).
2. Click the Calculate button to get the result.

What is an equilateral triangle?

An equilateral triangle is a special kind of triangle whose definition is based on the equality of sides and angles. Specifically:

1. Equality of sides: all three sides of an equilateral triangle have the same length. This means that if you measure any two sides, they all have the same length.

2. Equality of angles : Since the sum of the interior angles of a triangle is 180 degrees, each interior angle of an equilateral triangle is 60 degrees. This is because three identical angles must add up to equal 180 degrees, so each angle is { \color{Orange} \frac{180^\circ}{3} = 60^\circ }.

3. Symmetry: The equilateral triangle has a variety of symmetries. It is not only an axially symmetric figure, i.e., symmetric about a line passing through each vertex and the midpoint of the opposite side, but also rotationally symmetric, i.e., it coincides with the original figure after being rotated 120 or 240 degrees about the center.

4. Height, midline, and angle bisector: in an equilateral triangle, the vertical line from each vertex to the opposite side (the height), the line segment connecting the vertex to the midpoint of the opposite side (the midline), and the angle bisector of each interior angle coincide.

5. Center of Gravity, Inner and Outer Concentricity: The center of gravity (the intersection of the three medians of a triangle), the inner center (the intersection of the triangle’s interior angle bisectors), and the outer center (the intersection of the triangle’s exterior angle bisectors) of an equilateral triangle are located in the same point, the center of the triangle.

6. Ratio of area to perimeter: The ratio of the area to the perimeter of an equilateral triangle is a fixed constant, independent of the length of the sides; this ratio is approximately 0.1702.

These properties of the equilateral triangle give it a special place in geometry, and it occurs widely in art, architecture, and nature.

Calculating the area of an equilateral triangle

The formula for calculating the area of an equilateral triangle is:

{ \color{Orange} S = \frac{\sqrt{3}}{4} a^2 }

Where :.

• { \color{Orange} S } is the area of the triangle
• { \color{Orange} a } is the length of the side of the equilateral triangle.

Steps for calculating the area of an equilateral triangle:

1. Determine the side lengths: first, you need to know the side lengths { \color{Orange} a } of the equilateral triangle.

2. Calculate the square of the side length: multiply the side length { \color{Orange} a } by itself, i.e. { \color{Orange} a^2 }.

3. Apply the formula: multiply the square of the side length { \color{Orange} \frac{\sqrt{3}}{4} }. This step involves calculating the value of { \color{Orange} \sqrt{3} } (i.e., the square root of 3), which is approximately equal to 1.732.

4. Deriving the area: After completing the above calculations, the value obtained is the area of the equilateral triangle.

For example, if the side lengths of an equilateral triangle are { \color{Orange} a = 6 } units, then its area is calculated as follows:

1. Calculate the square of the side length: { \color{Orange} 6^2 = 36 }.

2. Apply the area formula: { \color{Orange} S = \frac{\sqrt{3}}{4} \times 36 }.

3. Calculate the value of { \color{Orange} \sqrt{3} }: { \color{Orange} \sqrt{3} \approx 1.732 }.

4. Calculate the area: { \color{Orange} S \approx \frac{1.732}{4} \times 36 \approx 0.433 \times 36 = 15.588 }.

So, the area of this equilateral triangle is approximately { \color{Orange} 15.588 } square units.

Calculating the perimeter of an equilateral triangle

The formula for calculating the perimeter of an equilateral triangle is:

{ \color{Orange} P = 3a }

• where { \color{Orange} P } is the perimeter of the triangle and { \color{Orange} a } is the length of the sides of the equilateral triangle.

Steps for calculating the perimeter of an equilateral triangle:

1. Determine the side lengths: first, you need to know the side lengths { \color{Orange} a } of the equilateral triangle.

2. Calculate the side length: multiply the side length { \color{Orange} a } by 3.

3. Derive the perimeter: After completing the above calculations, the resulting value is the perimeter of the equilateral triangle.

For example, if the side length of an equilateral triangle is { \color{Orange} a = 6 } units, then its perimeter is calculated as follows:

1. Calculate the side length: { \color{Orange} 3 \times 6 = 18 }.

2. Find the perimeter: { \color{Orange} P = 18 }.

So, the perimeter of this equilateral triangle is { \color{Orange} 18 } units.

Calculating the height of an equilateral triangle

The formula for calculating the height of an equilateral triangle is:

{ \color{Orange} h = \frac{a \sqrt{3}}{2} }

• where { \color{Orange} h } is the height of the equilateral triangle and { \color{Orange} a } is the length of the side of the equilateral triangle.

Steps for calculating the height of an equilateral triangle:

1. Determine the side lengths: first, you need to know the side lengths { \color{Orange} a } of the equilateral triangle.

2. Calculate the square of the side length: multiply the side length { \color{Orange} a } by itself, i.e. { \color{Orange} a^2 }.

3. Apply the formula: divide the square of the side length by 2.

4. Derive the altitude: after completing the above calculations, the value obtained is the area of the equilateral triangle.

For example, if the side length of an equilateral triangle is { \color{Orange} a = 6 } units, then its altitude is calculated as follows:

1. Calculate the square of the side length: { \color{Orange} 6^2 = 36 }.

2. Apply the area formula: { \color{Orange} h = \frac{36}{2} }.

3. Calculate the value of { \color{Orange} \sqrt{3} }: { \color{Orange} \sqrt{3} \approx 1.732 }.

4. Calculate the height: { \color{Orange} h \approx \frac{1.732}{2} \times 6 = 0.866 \times 6 = 5.196 }.

So, the area of this equilateral triangle is about { \color{Orange} 5.196 } units.

More on equilateral triangles:

• Wikipedia on equilateral triangles

This article is also available in the following languages: 简体中文 (Chinese (Simplified)) English 日本語 (Japanese) 한국어 (Korean) Русский (Russian) Deutsch (German)