Triangle calculator

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

You have entered side a, b and angle β.

Triangle has two solutions: a=38.9; b=32.1; c=8.74332847182 and a=38.9; b=32.1; c=55.2219521674.

#1 Obtuse scalene triangle.

Sides: a = 38.9   b = 32.1   c = 8.74332847182

Area: T = 96.81099040256
Perimeter: p = 79.74332847182
Semiperimeter: s = 39.87216423591

Angle ∠ A = α = 136.3879819764° = 136°22'47″ = 2.38802768882 rad
Angle ∠ B = β = 34.7° = 34°42' = 0.60656292504 rad
Angle ∠ C = γ = 8.92201802363° = 8°55'13″ = 0.1565686515 rad

Height: ha = 4.97773729576
Height: hb = 6.03217697212
Height: hc = 22.14549734615

Median: ma = 13.23334807905
Median: mb = 23.17881149758
Median: mc = 35.39334844722

Inradius: r = 2.42880390347
Circumradius: R = 28.19435311906

Vertex coordinates: A[8.74332847182; 0] B[0; 0] C[31.98114031961; 22.14549734615]
Centroid: CG[13.57548959714; 7.38216578205]
Coordinates of the circumscribed circle: U[4.37216423591; 27.85325392752]
Coordinates of the inscribed circle: I[7.77216423591; 2.42880390347]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 43.62201802363° = 43°37'13″ = 2.38802768882 rad
∠ B' = β' = 145.3° = 145°18' = 0.60656292504 rad
∠ C' = γ' = 171.0879819764° = 171°4'47″ = 0.1565686515 rad




How did we calculate this triangle?

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 = 38.9 ; ; b = 32.1 ; ; c = 8.74 ; ;

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

p = a+b+c = 38.9+32.1+8.74 = 79.74 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 79.74 }{ 2 } = 39.87 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.87 * (39.87-38.9)(39.87-32.1)(39.87-8.74) } ; ; T = sqrt{ 9372.16 } = 96.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 96.81 }{ 38.9 } = 4.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 96.81 }{ 32.1 } = 6.03 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 96.81 }{ 8.74 } = 22.14 ; ;

5. 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{ 38.9**2-32.1**2-8.74**2 }{ 2 * 32.1 * 8.74 } ) = 136° 22'47" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32.1**2-38.9**2-8.74**2 }{ 2 * 38.9 * 8.74 } ) = 34° 42' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.74**2-38.9**2-32.1**2 }{ 2 * 32.1 * 38.9 } ) = 8° 55'13" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 96.81 }{ 39.87 } = 2.43 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 38.9 }{ 2 * sin 136° 22'47" } = 28.19 ; ;





#2 Obtuse scalene triangle.

Sides: a = 38.9   b = 32.1   c = 55.2219521674

Area: T = 611.4177421012
Perimeter: p = 126.2219521674
Semiperimeter: s = 63.1109760837

Angle ∠ A = α = 43.62201802363° = 43°37'13″ = 0.76113157654 rad
Angle ∠ B = β = 34.7° = 34°42' = 0.60656292504 rad
Angle ∠ C = γ = 101.6879819764° = 101°40'47″ = 1.77546476377 rad

Height: ha = 31.43553429826
Height: hb = 38.09545433652
Height: hc = 22.14549734615

Median: ma = 40.76215049642
Median: mb = 44.98444449443
Median: mc = 22.57223527025

Inradius: r = 9.68881593735
Circumradius: R = 28.19435311906

Vertex coordinates: A[55.2219521674; 0] B[0; 0] C[31.98114031961; 22.14549734615]
Centroid: CG[29.06769749567; 7.38216578205]
Coordinates of the circumscribed circle: U[27.6109760837; -5.70875658138]
Coordinates of the inscribed circle: I[31.0109760837; 9.68881593735]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.3879819764° = 136°22'47″ = 0.76113157654 rad
∠ B' = β' = 145.3° = 145°18' = 0.60656292504 rad
∠ C' = γ' = 78.32201802363° = 78°19'13″ = 1.77546476377 rad

Calculate another triangle

How did we calculate this triangle?

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 = 38.9 ; ; b = 32.1 ; ; c = 55.22 ; ;

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

p = a+b+c = 38.9+32.1+55.22 = 126.22 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 126.22 }{ 2 } = 63.11 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 63.11 * (63.11-38.9)(63.11-32.1)(63.11-55.22) } ; ; T = sqrt{ 373831.26 } = 611.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 611.42 }{ 38.9 } = 31.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 611.42 }{ 32.1 } = 38.09 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 611.42 }{ 55.22 } = 22.14 ; ;

5. 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{ 38.9**2-32.1**2-55.22**2 }{ 2 * 32.1 * 55.22 } ) = 43° 37'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32.1**2-38.9**2-55.22**2 }{ 2 * 38.9 * 55.22 } ) = 34° 42' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 55.22**2-38.9**2-32.1**2 }{ 2 * 32.1 * 38.9 } ) = 101° 40'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 611.42 }{ 63.11 } = 9.69 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 38.9 }{ 2 * sin 43° 37'13" } = 28.19 ; ;




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