45-45-90 Triangle Calculator

What is a 45-45-90 Triangle?

A 45-45-90 triangle, also known as an isosceles right triangle, is a special right triangle with two 45° angles and one 90° angle. This triangle has unique properties that make it valuable in geometry, trigonometry, and real-world applications.

Key Properties of 45-45-90 Triangles

• Two equal sides (legs) opposite the 45° angles
• The hypotenuse is opposite the 90° angle
• The ratio of leg to hypotenuse is always 1 : √2
• The triangle is isosceles (two equal sides) and right-angled

Formulas for 45-45-90 Triangles

Let \(a\) be the length of a leg. Then:

1. Leg: \(a = b\)
2. Hypotenuse: \(c = a\sqrt{2}\)
3. Height (from right angle to hypotenuse): \(h = \frac{a\sqrt{2}}{2}\)
4. Area: \(A = \frac{a^2}{2}\)
5. Perimeter: \(P = a(2 + \sqrt{2})\)
6. Inradius: \(r = a(\sqrt{2} - 1)\)
7. Circumradius: \(R = \frac{a\sqrt{2}}{2}\)

Step-by-Step Calculations

Let's calculate these properties for a 45-45-90 triangle with leg \(a = 5\) units:

1. Leg: \[a = b = 5 \text{ units}\]
2. Hypotenuse: \[c = a\sqrt{2} = 5\sqrt{2} \approx 7.07 \text{ units}\]
3. Height: \[h = \frac{a\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \approx 3.54 \text{ units}\]
4. Area: \[A = \frac{a^2}{2} = \frac{5^2}{2} = 12.5 \text{ square units}\]
5. Perimeter: \[P = a(2 + \sqrt{2}) = 5(2 + \sqrt{2}) \approx 17.07 \text{ units}\]
6. Inradius: \[r = a(\sqrt{2} - 1) = 5(\sqrt{2} - 1) \approx 2.07 \text{ units}\]
7. Circumradius: \[R = \frac{a\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \approx 3.54 \text{ units}\]

Visual Representation

This diagram illustrates a 45-45-90 triangle with the calculated dimensions and angles.

Applications

45-45-90 triangles are commonly used in various fields:

• Architecture: For designing symmetrical structures and roof pitches
• Engineering: In machine design and structural analysis
• Computer Graphics: For creating isometric projections
• Navigation: In compass rose designs and direction finding

Example Problem

A square has a diagonal of 10 units. Find the side length of the square and its area.

Solution:

1. The diagonal of a square forms two 45-45-90 triangles
2. The diagonal is the hypotenuse (c) of these triangles: \(c = 10\) units
3. To find the side (a), use: \(a = \frac{c}{\sqrt{2}}\) \[a = \frac{10}{\sqrt{2}} \approx 7.07 \text{ units}\]
4. The area of the square is: \[A = a^2 = 7.07^2 \approx 50 \text{ square units}\]

This example demonstrates how understanding 45-45-90 triangles can help solve problems involving squares and other geometric shapes.