15-75-90 Triangle Calculator

What is a 15-75-90 Triangle?

A 15-75-90 triangle is a right-angled triangle where the angles are exactly 15°, 75°, and 90°. This type of triangle is a special case of a right triangle and has specific side length ratios that make it distinct from other right triangles.

The triangle gets its name from its three angles: 15 degrees, 75 degrees, and the right angle of 90 degrees. The sides opposite these angles follow a specific ratio that can be derived using trigonometric relationships.

This triangle is significant in geometry because it demonstrates how trigonometric identities can be used to find exact values for trigonometric functions at specific angles. The exact values for sine, cosine, and tangent at 15° and 75° can be derived from the side ratios of this triangle.

Properties of a 15-75-90 Triangle

The 15-75-90 triangle has several important properties that distinguish it from other right triangles:

Side Length Ratios

The sides of a 15-75-90 triangle follow a specific ratio based on the exact values of trigonometric functions at 15° and 75°:

• Side opposite 15°: 1 unit
• Side opposite 75°: √6 - √2 ≈ 0.5176 units
• Hypotenuse: √6 + √2 ≈ 2.934 units

Trigonometric Values

The exact trigonometric values for 15° and 75° can be derived from the side ratios:

Area and Perimeter

For a 15-75-90 triangle with side lengths based on the ratio 1 : √6 - √2 : √6 + √2:

• Area = (1 × (√6 - √2))/2 ≈ 0.2588 square units
• Perimeter = 1 + (√6 - √2) + (√6 + √2) = 2 + 2√6 ≈ 5.868 units

Using the Calculator

The 15-75-90 Triangle Calculator allows you to calculate various properties of a 15-75-90 triangle based on the side length you provide. Here's how to use it:

Input Options

You can specify any one of the three sides of the triangle:

• Side opposite 15° angle
• Side opposite 75° angle
• Hypotenuse (opposite 90° angle)

Calculation Process

1. Enter the length of one side in the input field
2. Select which side you're providing
3. Click "Calculate" to compute the triangle properties
4. View the results including all side lengths, area, and perimeter

Result Interpretation

The calculator provides:

• All three side lengths (scaled according to the 1:√6-√2:√6+√2 ratio)
• The area of the triangle
• The perimeter of the triangle
• A visual representation of the triangle

Worked Examples

Let's look at two examples of how to use the 15-75-90 Triangle Calculator.

Example 1: Given the side opposite 15°

Suppose we have a 15-75-90 triangle where the side opposite the 15° angle is 5 units.

Using the calculator:

1. Enter 5 in the "Side opposite 15° angle" field
2. Select "Side opposite 15° angle" from the dropdown
3. Click "Calculate"

The calculator will return:

• Side opposite 15°: 5 units
• Side opposite 75°: 5 × (√6 - √2) ≈ 2.588 units
• Hypotenuse: 5 × (√6 + √2) ≈ 14.67 units
• Area: (5 × 2.588)/2 ≈ 6.47 square units
• Perimeter: 5 + 2.588 + 14.67 ≈ 22.26 units

Example 2: Given the hypotenuse

Now consider a 15-75-90 triangle with a hypotenuse of 10 units.

Using the calculator:

1. Enter 10 in the "Hypotenuse" field
2. Select "Hypotenuse" from the dropdown
3. Click "Calculate"

The calculator will return:

• Side opposite 15°: 10 / (√6 + √2) ≈ 3.415 units
• Side opposite 75°: 10 × (√6 - √2) / (√6 + √2) ≈ 1.707 units
• Hypotenuse: 10 units
• Area: (3.415 × 1.707)/2 ≈ 2.934 square units
• Perimeter: 3.415 + 1.707 + 10 ≈ 15.122 units