Triangle calculator

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

You have entered side c, angle α and angle γ.

Obtuse scalene triangle.

Sides: a = 30.95552980808   b = 72.13436967816   c = 60

Area: T = 914.5510526403
Perimeter: p = 163.0898994862
Semiperimeter: s = 81.54444974312

Angle ∠ A = α = 25° = 0.4366332313 rad
Angle ∠ B = β = 100° = 1.7455329252 rad
Angle ∠ C = γ = 55° = 0.96599310886 rad

Height: ha = 59.08884651807
Height: hb = 25.35770957044
Height: hc = 30.48550175468

Median: ma = 64.51441650017
Median: mb = 31.27877506671
Median: mc = 46.6998504744

Inradius: r = 11.21553554834
Circumradius: R = 36.62332376628

Vertex coordinates: A[60; 0] B[0; 0] C[-5.37553311009; 30.48550175468]
Centroid: CG[18.20882229664; 10.16216725156]
Coordinates of the circumscribed circle: U[30; 21.00662261463]
Coordinates of the inscribed circle: I[9.41108006496; 11.21553554834]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155° = 0.4366332313 rad
∠ B' = β' = 80° = 1.7455329252 rad
∠ C' = γ' = 125° = 0.96599310886 rad

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How did we calculate this triangle?

1. Input data entered: side c, angle α and angle γ.

c = 60 ; ; alpha = 25° ; ; gamma = 55° ; ;

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

 fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 60 * fraction{ sin 25° }{ sin 55° } = 30.96 ; ;

3. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 30.96**2+60**2 - 2 * 30.96 * 60 * cos 100° } ; ; b = 72.13 ; ;
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 = 30.96 ; ; b = 72.13 ; ; c = 60 ; ;

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

p = a+b+c = 30.96+72.13+60 = 163.09 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 163.09 }{ 2 } = 81.54 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 81.54 * (81.54-30.96)(81.54-72.13)(81.54-60) } ; ; T = sqrt{ 836402.67 } = 914.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 914.55 }{ 30.96 } = 59.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 914.55 }{ 72.13 } = 25.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 914.55 }{ 60 } = 30.49 ; ;

8. 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{ 72.13**2+60**2-30.96**2 }{ 2 * 72.13 * 60 } ) = 25° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 30.96**2+60**2-72.13**2 }{ 2 * 30.96 * 60 } ) = 100° ; ; gamma = 180° - alpha - beta = 180° - 25° - 100° = 55° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 914.55 }{ 81.54 } = 11.22 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 30.96 }{ 2 * sin 25° } = 36.62 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 72.13**2+2 * 60**2 - 30.96**2 } }{ 2 } = 64.514 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 30.96**2 - 72.13**2 } }{ 2 } = 31.278 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 72.13**2+2 * 30.96**2 - 60**2 } }{ 2 } = 46.699 ; ;
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