Select the type of special triangle and input known values to calculate unknown sides and angles.
30-60-90 Triangle
45-45-90 Triangle
Understanding Special Triangles
Special triangles are right-angled triangles with specific angle measures that lead to predictable relationships between their side lengths. Knowing these relationships allows us to quickly calculate unknown sides or angles if we know at least one side. The two most common special triangles are the 30-60-90 triangle and the 45-45-90 triangle.
1. The 30-60-90 Triangle
A 30-60-90 triangle is a right triangle whose angles measure 30 degrees, 60 degrees, and 90 degrees. The sides of this triangle have a fixed ratio:
The side opposite the 30° angle (the shortest side) is represented as x.
The side opposite the 60° angle is x√3 (x times the square root of 3).
The side opposite the 90° angle (the hypotenuse) is 2x (twice the shortest side).
This means:
Hypotenuse = 2 * (Shortest Side)
Side Opposite 60° = Shortest Side * √3
If you know any one side, you can find the other two. For example, if the shortest side is 5, the hypotenuse is 10, and the side opposite 60° is 5√3 (approximately 8.66). If the hypotenuse is 12, the shortest side is 6, and the side opposite 60° is 6√3 (approximately 10.39).
2. The 45-45-90 Triangle
A 45-45-90 triangle is a right triangle whose angles measure 45 degrees, 45 degrees, and 90 degrees. This is an isosceles right triangle, meaning two of its sides are equal in length. The sides of this triangle also have a fixed ratio:
The two equal sides (opposite the 45° angles) are represented as x.
The hypotenuse (opposite the 90° angle) is x√2 (x times the square root of 2).
This means:
Hypotenuse = (Leg Length) * √2
Leg Length = Hypotenuse / √2
If you know one of the equal sides (legs), say 7, then the hypotenuse is 7√2 (approximately 9.90). If you know the hypotenuse is 10, then each leg is 10/√2 (approximately 7.07).
Use Cases
Special triangles appear frequently in geometry, trigonometry, physics, engineering, and architecture. They simplify calculations involving right angles, especially when dealing with:
Coordinate Geometry: Calculating distances or properties of points.
Trigonometry: Deriving basic trigonometric values (sin, cos, tan) for common angles.
Construction and Design: Designing structures or components that require specific angles and proportions.
Physics: Analyzing forces, vectors, or motion involving specific angular relationships.
This calculator helps quickly determine missing side lengths based on the properties of these fundamental geometric shapes.
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