Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, angle β and angle γ.

Right scalene triangle.

Sides: a = 2.2   b = 1.27701705922   c = 2.54403411844

Area: T = 1.39771876514
Perimeter: p = 6.01105117767
Semiperimeter: s = 3.00552558883

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 1.27701705922
Height: hb = 2.2
Height: hc = 1.1

Median: ma = 1.68802777548
Median: mb = 2.29898325994
Median: mc = 1.27701705922

Inradius: r = 0.46549147039
Circumradius: R = 1.27701705922

Vertex coordinates: A[2.54403411844; 0] B[0; 0] C[1.90552558883; 1.1]
Centroid: CG[1.48218656909; 0.36766666667]
Coordinates of the circumscribed circle: U[1.27701705922; -0]
Coordinates of the inscribed circle: I[1.73550852961; 0.46549147039]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: side a, angle β and angle γ.

a = 2.2 ; ; beta = 30° ; ; gamma = 90° ; ;

2. From angle β and angle γ we calculate angle α:

 beta + gamma + alpha = 180° ; ; alpha = 180° - beta - gamma = 180° - 30 ° - 90 ° = 60 ° ; ;

3. From angle β, angle α and side a we calculate side b - By using the law of sines, we calculate unknown side b:

 fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 2.2 * fraction{ sin 30° }{ sin 60° } = 1.27 ; ;

4. From angle γ, angle α and side a we calculate side c - By using the law of sines, we calculate unknown side c:

 fraction{ c }{ a } = fraction{ sin gamma }{ sin alpha } ; ; ; ; c = a * fraction{ sin gamma }{ sin alpha } ; ; ; ; c = 2.2 * fraction{ sin 90° }{ sin 60° } = 2.54 ; ;
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 2.2 ; ; b = 1.27 ; ; c = 2.54 ; ;

5. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 2.2+1.27+2.54 = 6.01 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 6.01 }{ 2 } = 3.01 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.01 * (3.01-2.2)(3.01-1.27)(3.01-2.54) } ; ; T = sqrt{ 1.95 } = 1.4 ; ;

8. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.4 }{ 2.2 } = 1.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.4 }{ 1.27 } = 2.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.4 }{ 2.54 } = 1.1 ; ;

9. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 1.27**2+2.54**2-2.2**2 }{ 2 * 1.27 * 2.54 } ) = 60° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.2**2+2.54**2-1.27**2 }{ 2 * 2.2 * 2.54 } ) = 30° ; ;
 gamma = 180° - alpha - beta = 180° - 60° - 30° = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.4 }{ 3.01 } = 0.46 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 2.2 }{ 2 * sin 60° } = 1.27 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.27**2+2 * 2.54**2 - 2.2**2 } }{ 2 } = 1.68 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.54**2+2 * 2.2**2 - 1.27**2 } }{ 2 } = 2.29 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.27**2+2 * 2.2**2 - 2.54**2 } }{ 2 } = 1.27 ; ;
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