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45-45-90 Triangle Calculator

45-45-90 Triangle Diagram

1. What is a 45-45-90 Triangle Calculator?

Definition: This calculator computes the sides, area, and perimeter of a 45-45-90 triangle given the length of one leg a.

Purpose: It is used in geometry and engineering to determine the properties of a 45-45-90 triangle, which is an isosceles right triangle with fixed side ratios.

2. How Does the Calculator Work?

The calculator uses the following formulas for a 45-45-90 triangle, where a is one of the legs:

  • Leg b: b=a (since the triangle is isosceles)
  • Hypotenuse c: c=a2
  • Area: Area=12a2
  • Perimeter: Perimeter=a+b+c=2a+a2

Unit Conversions:

  • Input Dimensions: m, cm (1 m = 100 cm), mm (1 m = 1000 mm), in (1 m = 39.3701 in), ft (1 m = 3.28084 ft), yd (1 m = 1.09361 yd)
  • Output Sides and Perimeter: m, cm, mm, in, ft, yd
  • Output Area: m², cm² (1 m² = 10000 cm²), mm² (1 m² = 1000000 mm²), in² (1 m² = 1550.0031 in²), ft² (1 m² = 10.7639 ft²), yd² (1 m² = 1.19599 yd²)

Steps:

  • Input the length of leg a with its unit.
  • Convert the dimension to meters for calculation.
  • Calculate b, c, area, and perimeter using the formulas.
  • Convert each result to its respective selected unit for display, using scientific notation for values less than 0.001, otherwise with 4 decimal places.

3. Importance of 45-45-90 Triangle Calculations

Calculating the properties of a 45-45-90 triangle is crucial for:

  • Geometry Education: Understanding special triangles and their properties.
  • Engineering Design: Using isosceles right triangles in structural components.
  • Trigonometry Applications: Simplifying calculations involving 45° angles.

4. Using the Calculator

Examples:

  • Example 1: For a triangle with a=10cm:
    • Convert: a=10/100=0.1m
    • Leg b: b=a=0.1m
    • Hypotenuse c: c=a2=0.1×2=0.1414m
    • Area: Area=12a2=12×0.12=0.005m2
    • Perimeter: Perimeter=2a+a2=2×0.1+0.1414=0.3414m
    • Convert: b=10cm, c=14.14cm, Area = 50cm2, Perimeter = 34.14cm
  • Example 2: For a triangle with a=2ft:
    • Convert: a=2/3.28084=0.6096m
    • Leg b: b=a=0.6096m
    • Hypotenuse c: c=a2=0.6096×2=0.8622m
    • Area: Area=12a2=12×0.60962=0.1858m2
    • Perimeter: Perimeter=2a+a2=2×0.6096+0.8622=2.0814m
    • Convert: b=2ft, c=2.8284ft, Area = 2ft2, Perimeter = 6.8284ft

5. Frequently Asked Questions (FAQ)

Q: What is a 45-45-90 triangle?
A: A 45-45-90 triangle is an isosceles right triangle with two 45° angles and one 90° angle, where the legs are equal, and the hypotenuse is 2 times the leg length.

Q: Why are 45-45-90 triangles important?
A: They have consistent side ratios, making them useful in trigonometry, geometry, and engineering for simplifying calculations.

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