Triangle calculator

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

You have entered side b, c and angle γ.

Acute scalene triangle.

Sides: a = 88.89224664176   b = 57   c = 78

Area: T = 2194.01993225
Perimeter: p = 223.8922466418
Semiperimeter: s = 111.9466233209

Angle ∠ A = α = 80.73882060879° = 80°44'18″ = 1.40991475284 rad
Angle ∠ B = β = 39.26217939121° = 39°15'42″ = 0.6855247574 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: ha = 49.36334480157
Height: hb = 76.98331341227
Height: hc = 56.2576905705

Median: ma = 51.87551612388
Median: mb = 78.61773345573
Median: mc = 63.67444477236

Inradius: r = 19.59988668811
Circumradius: R = 45.03333209968

Vertex coordinates: A[78; 0] B[0; 0] C[68.82660934987; 56.2576905705]
Centroid: CG[48.94220311662; 18.75223019017]
Coordinates of the circumscribed circle: U[39; 22.51766604984]
Coordinates of the inscribed circle: I[54.94662332088; 19.59988668811]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 99.26217939121° = 99°15'42″ = 1.40991475284 rad
∠ B' = β' = 140.7388206088° = 140°44'18″ = 0.6855247574 rad
∠ C' = γ' = 120° = 1.04771975512 rad

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How did we calculate this triangle?

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

b = 57 ; ; c = 78 ; ; gamma = 60° ; ;

2. From angle γ, side b and c we calculate a - by using the law of cosines and quadratic equation:

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 78**2 = 57**2 + a**2 - 2 * 57 * a * cos(60° ) ; ; ; ; ; ; a**2 -57a -2835 =0 ; ; a=1; b=-57; c=-2835 ; ; D = b**2 - 4ac = 57**2 - 4 * 1 * (-2835) = 14589 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 57 ± sqrt{ 14589 } }{ 2 } ; ; a_{1,2} = 28.5 ± 60.3924664176 ; ; a_{1} = 88.8924664176 ; ; a_{2} = -31.8924664176 ; ; ; ; (a -88.8924664176) (a +31.8924664176) = 0 ; ;
 ; ; a > 0 ; ; ; ; a = 88.892 ; ;


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 = 88.89 ; ; b = 57 ; ; c = 78 ; ;

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

p = a+b+c = 88.89+57+78 = 223.89 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 223.89 }{ 2 } = 111.95 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 111.95 * (111.95-88.89)(111.95-57)(111.95-78) } ; ; T = sqrt{ 4813720.79 } = 2194.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2194.02 }{ 88.89 } = 49.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2194.02 }{ 57 } = 76.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2194.02 }{ 78 } = 56.26 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 88.89**2-57**2-78**2 }{ 2 * 57 * 78 } ) = 80° 44'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 57**2-88.89**2-78**2 }{ 2 * 88.89 * 78 } ) = 39° 15'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 78**2-88.89**2-57**2 }{ 2 * 57 * 88.89 } ) = 60° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2194.02 }{ 111.95 } = 19.6 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 88.89 }{ 2 * sin 80° 44'18" } = 45.03 ; ;




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