Triangle calculator

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

You have entered side b, c and angle α.

Acute scalene triangle.

Sides: a = 104.0966109437   b = 66   c = 120

Area: T = 3429.461059899
Perimeter: p = 290.0966109437
Semiperimeter: s = 145.0488054719

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 33.30443051802° = 33°18'16″ = 0.58112697805 rad
Angle ∠ C = γ = 86.69656948198° = 86°41'44″ = 1.51331253219 rad

Height: ha = 65.89902742383
Height: hb = 103.9233048454
Height: hc = 57.15876766498

Median: ma = 81.66439455329
Median: mb = 107.3733181009
Median: mc = 63.21439225171

Inradius: r = 23.6443616632
Circumradius: R = 60.10999168053

Vertex coordinates: A[120; 0] B[0; 0] C[87; 57.15876766498]
Centroid: CG[69; 19.05325588833]
Coordinates of the circumscribed circle: U[60; 3.46441016151]
Coordinates of the inscribed circle: I[79.04880547187; 23.6443616632]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 146.696569482° = 146°41'44″ = 0.58112697805 rad
∠ C' = γ' = 93.30443051802° = 93°18'16″ = 1.51331253219 rad

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How did we calculate this triangle?

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

b = 66 ; ; c = 120 ; ; alpha = 60° ; ;

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{ 66**2+120**2 - 2 * 66 * 120 * cos(60° ) } ; ; a = 104.1 ; ;


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 = 104.1 ; ; b = 66 ; ; c = 120 ; ;

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

p = a+b+c = 104.1+66+120 = 290.1 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 290.1 }{ 2 } = 145.05 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 145.05 * (145.05-104.1)(145.05-66)(145.05-120) } ; ; T = sqrt{ 11761200 } = 3429.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3429.46 }{ 104.1 } = 65.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3429.46 }{ 66 } = 103.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3429.46 }{ 120 } = 57.16 ; ;

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{ 104.1**2-66**2-120**2 }{ 2 * 66 * 120 } ) = 60° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 66**2-104.1**2-120**2 }{ 2 * 104.1 * 120 } ) = 33° 18'16" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 120**2-104.1**2-66**2 }{ 2 * 66 * 104.1 } ) = 86° 41'44" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3429.46 }{ 145.05 } = 23.64 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 104.1 }{ 2 * sin 60° } = 60.1 ; ;




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