Isosceles Triangle Calculator

Data Source & Methodology

An isosceles triangle is defined as a triangle with *at least* two equal sides (the "legs"). This also means it has two equal base angles. The calculations for its properties are derived from fundamental geometric and trigonometric principles.

• Authoritative Source: The methods are based on **Euclid's *Elements*** (specifically Book I, Proposition 5), the **Pythagorean Theorem**, and the **Law of Sines** and **Law of Cosines**.
• Methodology: "Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte." The core strategy involves bisecting the isosceles triangle's vertex angle ($\alpha$) to form two congruent 90° right triangles. The properties of these right triangles are then used to find all other values.

The Formulas Explained

All calculations are based on the right triangle formed by the height ($h$), half the base ($b/2$), and one of the equal sides ($a$).

Key Relationships

These formulas are the building blocks for the calculator.

Pythagorean Theorem:

Trigonometry (SOH CAH TOA):

Angle Sum:

Core Calculation Formulas

Once the base properties ($a$, $b$, $h$) are found, the area and perimeter are simple.

Area ($A$):

Perimeter ($P$):

Glossary of Variables

• Equal Side ($a$): The length of one of the two equal sides (also called "legs").
• Base ($b$): The length of the third, unequal side.
• Height ($h$): The altitude of the triangle, drawn from the vertex angle perpendicular to the base.
• Base Angle ($\beta$): One of the two equal angles opposite the equal sides.
• Vertex Angle ($\alpha$): The angle formed between the two equal sides ($a$).
• Area ($A$): The total space enclosed by the triangle.
• Perimeter ($P$): The total distance around the triangle's boundary.

How It Works: A Step-by-Step Example

Let's solve a triangle where you only know the **base ($b$)** and the **vertex angle ($\alpha$)**.

Problem: You have an isosceles triangle with a **base of 10 in** and a **vertex angle of 30°**. Find its other properties.

1. Enter Known Values:

   In the calculator, you would enter "10" for Base (b) and "30" for Vertex Angle (α). Press "Calculate".

2. Step 1: Find Base Angles ($\beta$):

   The calculator first finds the base angles using the angle sum formula.

   $$ \beta = (180^\circ - \alpha) / 2 = (180^\circ - 30^\circ) / 2 = 75^\circ $$
3. Step 2: Find Height ($h$):

   Using the right triangle formed by $h$ and $b/2$ (which is 5), we use the tangent function.

   $$ \tan(\beta) = h / (b/2) \implies h = (b/2) \tan(\beta) $$ $$ h = 5 \times \tan(75^\circ) \approx 5 \times 3.732 = 18.66 \text{ in} $$
4. Step 3: Find Equal Side ($a$):

   We can use the cosine function (or Pythagorean theorem).

   $$ \cos(\beta) = (b/2) / a \implies a = (b/2) / \cos(\beta) $$ $$ a = 5 / \cos(75^\circ) \approx 5 / 0.2588 = 19.32 \text{ in} $$
5. Step 4: Find Area and Perimeter:

   Now the calculator has all base properties to solve the rest.

   $$ A = (b \times h) / 2 = (10 \times 18.66) / 2 = 93.3 \text{ in}^2 $$ $$ P = 2a + b = (2 \times 19.32) + 10 = 48.64 \text{ in} $$

Result: The calculator would fill in all remaining fields with these computed values.

Frequently Asked Questions (FAQ)