Triangle calculator

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

You have entered side b, c and angle α.

Obtuse scalene triangle.

Sides: a = 31.87546171439   b = 54   c = 67

Area: T = 849.274405708
Perimeter: p = 152.8754617144
Semiperimeter: s = 76.4377308572

Angle ∠ A = α = 28° = 0.48986921906 rad
Angle ∠ B = β = 52.68882898938° = 52°41'18″ = 0.92195841359 rad
Angle ∠ C = γ = 99.31217101062° = 99°18'42″ = 1.73333163271 rad

Height: ha = 53.288842403
Height: hb = 31.45545947067
Height: hc = 25.35114643904

Median: ma = 58.72439490794
Median: mb = 44.98332814392
Median: mc = 29.04772995137

Inradius: r = 11.11107268551
Circumradius: R = 33.94773353346

Vertex coordinates: A[67; 0] B[0; 0] C[19.32108299856; 25.35114643904]
Centroid: CG[28.77436099952; 8.45504881301]
Coordinates of the circumscribed circle: U[33.5; -5.4932865948]
Coordinates of the inscribed circle: I[22.4377308572; 11.11107268551]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152° = 0.48986921906 rad
∠ B' = β' = 127.3121710106° = 127°18'42″ = 0.92195841359 rad
∠ C' = γ' = 80.68882898938° = 80°41'18″ = 1.73333163271 rad

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How did we calculate this triangle?

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

b = 54 ; ; c = 67 ; ; alpha = 28° ; ;

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

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 54**2+67**2 - 2 * 54 * 67 * cos 28° } ; ; a = 31.87 ; ;


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 = 31.87 ; ; b = 54 ; ; c = 67 ; ;

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

p = a+b+c = 31.87+54+67 = 152.87 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 152.87 }{ 2 } = 76.44 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 76.44 * (76.44-31.87)(76.44-54)(76.44-67) } ; ; T = sqrt{ 721266.42 } = 849.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 849.27 }{ 31.87 } = 53.29 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 849.27 }{ 54 } = 31.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 849.27 }{ 67 } = 25.35 ; ;

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{ 54**2+67**2-31.87**2 }{ 2 * 54 * 67 } ) = 28° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 31.87**2+67**2-54**2 }{ 2 * 31.87 * 67 } ) = 52° 41'18" ; ; gamma = 180° - alpha - beta = 180° - 28° - 52° 41'18" = 99° 18'42" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 849.27 }{ 76.44 } = 11.11 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 31.87 }{ 2 * sin 28° } = 33.95 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 54**2+2 * 67**2 - 31.87**2 } }{ 2 } = 58.724 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 67**2+2 * 31.87**2 - 54**2 } }{ 2 } = 44.983 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 54**2+2 * 31.87**2 - 67**2 } }{ 2 } = 29.047 ; ;
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