Isosceles Triangle Calculator

Instantly solve for any property of an isosceles triangle from two known values.

Select the type of values you know, input them into the fields, and the calculator will find the rest.

Practical Examples

Click any example to automatically load the values into the calculator.

Calculate from Equal Side and Base

A_B

A triangle with equal sides of 10 and a base of 12.

sideA: 10

baseB: 12

Calculate from Base and Height

B_H

A triangle with a base of 6 and a height of 4.

baseB: 6

height: 4

Calculate from Side and Apex Angle

A_APEX

A triangle with equal sides of 8 and an apex angle of 30°.

sideA: 8

apexAngle: 30

Calculate from Base and Base Angle

B_BASE

A triangle with a base of 15 and a base angle of 75°.

baseB: 15

baseAngle: 75

Other Titles
Understanding the Isosceles Triangle: A Comprehensive Guide
Explore the properties, formulas, and applications of isosceles triangles with this detailed guide and our intuitive calculator.

What is an Isosceles Triangle?

  • Definition and Core Properties
  • Key Terminology Explained
  • Symmetry and the Altitude
An isosceles triangle is a cornerstone of geometry, defined as a triangle possessing at least two sides of equal length. This primary characteristic gives rise to another key property: the angles opposite the equal sides are also equal. The two equal sides are called 'legs,' the third side is the 'base,' the angle between the legs is the 'apex angle,' and the two equal angles are the 'base angles.'
The Importance of Symmetry
A critical feature of an isosceles triangle is its axis of symmetry. The altitude (height) drawn from the apex to the base not only meets the base at a right angle (90°) but also bisects it into two equal segments. This same altitude also bisects the apex angle. This perfect symmetry is incredibly useful as it divides the isosceles triangle into two identical right-angled triangles, simplifying calculations significantly.

Fundamental Examples

  • A triangle with side lengths 8, 8, 10 is an isosceles triangle.
  • A triangle with angles 50°, 50°, 80° is an isosceles triangle.
  • An equilateral triangle (all sides equal) is a special, more symmetric type of isosceles triangle.

Step-by-Step Guide to Using the Calculator

  • Selecting Your Calculation Method
  • Entering Your Known Values
  • Interpreting the Results
Our calculator is designed to be flexible, enabling you to solve for the triangle's properties from various combinations of known data.
How to Use:
1. Select Calculation Type: Begin by using the dropdown menu to choose the pair of values you already know (e.g., 'Equal Side (a) and Base (b)').
2. Enter Values: The appropriate input fields will appear. Enter your measurements here. Note that all angle inputs should be in degrees.
3. Calculate: Press the 'Calculate' button to see the magic happen.
4. Review Results: The results card will populate with all properties of the triangle, providing a complete overview.

Calculation Scenarios

  • Known: Equal Side a = 13, Base b = 10. Result: Height h = 12, Area = 60, Perimeter = 36.
  • Known: Base b = 16, Height h = 6. Result: Equal Side a = 10, Area = 48, Perimeter = 36.

Mathematical Formulas and Derivations

  • Calculating Height and Area
  • Finding the Perimeter and Angles
  • The Role of Trigonometry
All calculations are grounded in the Pythagorean theorem and fundamental trigonometric identities, which are applied to the two right-angled triangles created by the altitude.
Core Formulas (given equal side 'a' and base 'b'):
Height (h): h = √(a² - (b/2)²). Derived from the Pythagorean theorem in one of the small right triangles.
Area: Area = (1/2) * b * h. The universal formula for a triangle's area.
Perimeter (P): P = 2a + b. The total length of the triangle's boundary.
Base Angle (β): β = arccos((b/2) / a). Found using the cosine definition in a right triangle.
Apex Angle (α): α = 180° - 2β. Based on the fact that a triangle's interior angles sum to 180°.

Applying the Formulas

  • For a=5, b=6: h = √(5² - 3²) = √16 = 4. The Area is (1/2)*6*4 = 12.
  • For b=8, h=3: a = √(3² + 4²) = √25 = 5. The Perimeter is 2*5 + 8 = 18.

Real-World Applications

  • Architecture and Structural Engineering
  • Art, Design, and Aesthetics
  • Physics and Navigation
The isosceles triangle is more than a mere geometric shape; its inherent symmetry makes it indispensable in countless practical applications.
Examples in the Wild:
Architecture & Engineering: The gables of roofs, structural trusses, and supports for bridges frequently use isosceles triangles to ensure stability and evenly distribute weight.
Art & Design: Artists leverage isosceles triangles to create balance, harmony, and a sense of visual perspective. They are common in logos, patterns, and compositions.
Physics & Optics: The path of light refracting through a prism is analyzed using the geometry of an isosceles triangle. They are also used in mechanics to resolve force vectors.

Practical Examples

  • The classic A-frame house design.
  • A bicycle frame.
  • The shape of a yield sign on the road.

Common Questions and Misconceptions

  • Is an Equilateral Triangle Isosceles?
  • Can an Isosceles Triangle be a Right Triangle?
  • Input Validation and Constraints
Equilateral vs. Isosceles
Yes, every equilateral triangle (all three sides equal) is also an isosceles triangle because it meets the condition of having at least two equal sides. However, not all isosceles triangles are equilateral.
Isosceles Right Triangles
Absolutely. An isosceles right triangle has a 90° angle. Since the base angles must be equal, they must both be 45°. This means the apex angle is the right angle, and the two legs are the equal sides.
Triangle Inequality Theorem
A common error is to input side lengths that cannot form a triangle. For an isosceles triangle, the base must be shorter than the sum of the two equal sides (b < 2a). Our calculator validates this to prevent logical errors.

Important Considerations

  • A triangle with angles 45°, 45°, 90° is an isosceles right triangle.
  • Side lengths 5, 5, 12 cannot form a triangle because 5+5 is not greater than 12.