Triangle calculator

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

You have entered side a, b and angle γ.

Obtuse scalene triangle.

Sides: a = 45   b = 120   c = 135.2798972197

Area: T = 2658.981093313
Perimeter: p = 300.2798972198
Semiperimeter: s = 150.1399486099

Angle ∠ A = α = 19.12327025496° = 19°7'22″ = 0.33437541214 rad
Angle ∠ B = β = 60.87772974504° = 60°52'38″ = 1.06325092802 rad
Angle ∠ C = γ = 100° = 1.7455329252 rad

Height: ha = 118.1776930361
Height: hb = 44.31663488855
Height: hc = 39.31110753274

Median: ma = 125.873275384
Median: mb = 81.01104941313
Median: mc = 60.3110860716

Inradius: r = 17.71100708296
Circumradius: R = 68.68329341989

Vertex coordinates: A[135.2798972197; 0] B[0; 0] C[21.90106702319; 39.31110753274]
Centroid: CG[52.39332141431; 13.10436917758]
Coordinates of the circumscribed circle: U[67.63994860987; -11.92766663605]
Coordinates of the inscribed circle: I[30.13994860987; 17.71100708296]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.877729745° = 160°52'38″ = 0.33437541214 rad
∠ B' = β' = 119.123270255° = 119°7'22″ = 1.06325092802 rad
∠ C' = γ' = 80° = 1.7455329252 rad

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How did we calculate this triangle?

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

a = 45 ; ; b = 120 ; ; gamma = 100° ; ;

2. Calculation of the third side c of the triangle using a Law of Cosines

c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 45**2+120**2 - 2 * 45 * 120 * cos 100° } ; ; c = 135.28 ; ;


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 = 45 ; ; b = 120 ; ; c = 135.28 ; ;

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

p = a+b+c = 45+120+135.28 = 300.28 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 300.28 }{ 2 } = 150.14 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 150.14 * (150.14-45)(150.14-120)(150.14-135.28) } ; ; T = sqrt{ 7070179.6 } = 2658.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2658.98 }{ 45 } = 118.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2658.98 }{ 120 } = 44.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2658.98 }{ 135.28 } = 39.31 ; ;

7. 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{ 120**2+135.28**2-45**2 }{ 2 * 120 * 135.28 } ) = 19° 7'22" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 45**2+135.28**2-120**2 }{ 2 * 45 * 135.28 } ) = 60° 52'38" ; ; gamma = 180° - alpha - beta = 180° - 19° 7'22" - 60° 52'38" = 100° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2658.98 }{ 150.14 } = 17.71 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 45 }{ 2 * sin 19° 7'22" } = 68.68 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 120**2+2 * 135.28**2 - 45**2 } }{ 2 } = 125.873 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 135.28**2+2 * 45**2 - 120**2 } }{ 2 } = 81.01 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 120**2+2 * 45**2 - 135.28**2 } }{ 2 } = 60.311 ; ;
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