Triangle calculator

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

You have entered side a, b and angle γ.

Acute scalene triangle.

Sides: a = 80   b = 104   c = 82.39772850214

Area: T = 3242.046593411
Perimeter: p = 266.3977285021
Semiperimeter: s = 133.1998642511

Angle ∠ A = α = 49.17109561152° = 49°10'15″ = 0.85881950806 rad
Angle ∠ B = β = 79.62990438848° = 79°37'45″ = 1.3989788996 rad
Angle ∠ C = γ = 51.2° = 51°12' = 0.8943608577 rad

Height: ha = 81.05111483529
Height: hb = 62.34770371945
Height: hc = 78.69330281325

Median: ma = 84.86884646347
Median: mb = 62.37551255666
Median: mc = 83.13304508305

Inradius: r = 24.34399322471
Circumradius: R = 52.86436411474

Vertex coordinates: A[82.39772850214; 0] B[0; 0] C[14.40216430777; 78.69330281325]
Centroid: CG[32.26663093664; 26.23110093775]
Coordinates of the circumscribed circle: U[41.19986425107; 33.12545590256]
Coordinates of the inscribed circle: I[29.19986425107; 24.34399322471]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.8299043885° = 130°49'45″ = 0.85881950806 rad
∠ B' = β' = 100.3710956115° = 100°22'15″ = 1.3989788996 rad
∠ C' = γ' = 128.8° = 128°48' = 0.8943608577 rad

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How did we calculate this triangle?

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

a = 80 ; ; b = 104 ; ; gamma = 51.2° ; ;

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{ 80**2+104**2 - 2 * 80 * 104 * cos 51° 12' } ; ; c = 82.4 ; ;


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 = 80 ; ; b = 104 ; ; c = 82.4 ; ;

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

p = a+b+c = 80+104+82.4 = 266.4 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 266.4 }{ 2 } = 133.2 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 133.2 * (133.2-80)(133.2-104)(133.2-82.4) } ; ; T = sqrt{ 10510861.84 } = 3242.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3242.05 }{ 80 } = 81.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3242.05 }{ 104 } = 62.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3242.05 }{ 82.4 } = 78.69 ; ;

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{ 104**2+82.4**2-80**2 }{ 2 * 104 * 82.4 } ) = 49° 10'15" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 80**2+82.4**2-104**2 }{ 2 * 80 * 82.4 } ) = 79° 37'45" ; ; gamma = 180° - alpha - beta = 180° - 49° 10'15" - 79° 37'45" = 51° 12' ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3242.05 }{ 133.2 } = 24.34 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 80 }{ 2 * sin 49° 10'15" } = 52.86 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 104**2+2 * 82.4**2 - 80**2 } }{ 2 } = 84.868 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 82.4**2+2 * 80**2 - 104**2 } }{ 2 } = 62.375 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 104**2+2 * 80**2 - 82.4**2 } }{ 2 } = 83.13 ; ;
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