Data Source & Methodology
The properties of an equilateral triangle are foundational principles of Euclidean geometry. An equilateral triangle is defined as a polygon with three equal-length sides and three equal internal angles of 60°.
- Authoritative Source: Euclid's *Elements*, Book I, Proposition 1. This ancient Greek mathematical treatise (circa 300 BC) provides the original construction and proof of the equilateral triangle's properties.
- Methodology: "Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte." The formulas used are derived directly from the Pythagorean theorem by bisecting the triangle into two 30-60-90 right triangles.
The Formulas Explained
All properties of an equilateral triangle can be found if you know just one of its values. The primary variable is the **side length ($a$)**.
1. Calculations from Side ($a$)
These are the base formulas from which all others are derived.
Area ($A$):
Height ($h$):
Perimeter ($P$):
2. Reverse Formulas (Solving for $a$)
This calculator uses the following formulas to find the side ($a$) if you provide a different property.
From Area ($A$):
From Height ($h$):
From Perimeter ($P$):
Glossary of Variables
- Side ($a$): The length of any of the three equal sides of the triangle.
- Height ($h$): The altitude, or perpendicular distance, from any side to the opposite vertex.
- Area ($A$): The total space enclosed by the three sides of the triangle.
- Perimeter ($P$): The total distance around the triangle ($a + a + a$).
- Angle ($A, B, C$): The three internal angles, all of which are 60°.
How It Works: A Step-by-Step Example
Let's walk through a common problem where you only know the area.
Problem: You have an equilateral triangle with an **Area (A) of 43.3 cm²**. Find its side length, height, and perimeter.
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Select the "Given" Value:
In the calculator, choose "Area (A)" from the dropdown and enter "43.3". Select "cm" as the unit.
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Step 1: Find Side ($a$) from Area ($A$):
The calculator first finds the side length using the reverse formula:
$$ a = \sqrt{\frac{4 \times 43.3}{\sqrt{3}}} = \sqrt{\frac{173.2}{1.732}} = \sqrt{100} = 10 \text{ cm} $$ -
Step 2: Calculate Other Properties from Side ($a$):
Now that $a = 10 \text{ cm}$, the calculator finds all other values.
Height ($h$):
$$ h = \frac{10 \times \sqrt{3}}{2} = 5 \times 1.732 = 8.66 \text{ cm} $$Perimeter ($P$):
$$ P = 3 \times 10 = 30 \text{ cm} $$
Result: A triangle with an area of 43.3 cm² has a side of 10 cm, a height of 8.66 cm, and a perimeter of 30 cm.
Frequently Asked Questions (FAQ)
What defines an equilateral triangle?
An equilateral triangle is a triangle in which all three sides have the same length. As a consequence, all three internal angles are also equal, at 60° each.
How do you find the height of an equilateral triangle?
The height (or altitude) is found by drawing a line from one vertex perpendicular to the opposite side. This line bisects the side and the vertex angle, creating two 30-60-90 right triangles. Using the Pythagorean theorem ($a^2 = b^2 + c^2$) on one of these smaller triangles gives you the formula $h = (a \sqrt{3}) / 2$.
What is a 30-60-90 triangle?
This is a special right triangle formed when you bisect an equilateral triangle. Its angles are 30°, 60°, and 90°. The sides are in a constant ratio of $1:\sqrt{3}:2$. Our height formula is a direct application of this ratio.
Is an equilateral triangle also an isosceles triangle?
Yes. An isosceles triangle is defined as having *at least* two equal sides. Since an equilateral triangle has three equal sides, it meets this definition. It is a special, more specific case of an isosceles triangle.
What is the inradius of an equilateral triangle?
The inradius (radius of an inscribed circle) is the circle that fits perfectly inside the triangle. Its radius $r$ is $r = h / 3$ or $r = a / (2\sqrt{3})$.
What is the circumradius of an equilateral triangle?
The circumradius (radius of a circumscribed circle) is the circle that passes through all three vertices. Its radius $R$ is $R = 2h / 3$ or $R = a / \sqrt{3}$. You'll notice that $R = 2r$.
Tool developed by Ugo Candido.
Geometric content reviewed by the CalcDomain Editorial Board for mathematical accuracy.
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