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=37.9; b=26.4; c=18.21774715364 and a=37.9; b=26.4; c=40.59901553638.

#1 Obtuse scalene triangle.

Sides: a = 37.9   b = 26.4   c = 18.21774715364

Area: T = 217.8166091035
Perimeter: p = 82.51774715364
Semiperimeter: s = 41.25987357682

Angle ∠ A = α = 115.0769991308° = 115°4'12″ = 2.00883502186 rad
Angle ∠ B = β = 39.12° = 39°7'12″ = 0.68327728034 rad
Angle ∠ C = γ = 25.81100086922° = 25°48'36″ = 0.45504696316 rad

Height: ha = 11.49442528251
Height: hb = 16.50112190178
Height: hc = 23.91328784254

Median: ma = 12.46325693414
Median: mb = 26.64440074799
Median: mc = 31.36442460886

Inradius: r = 5.27992720615
Circumradius: R = 20.92109444007

Vertex coordinates: A[18.21774715364; 0] B[0; 0] C[29.40438134501; 23.91328784254]
Centroid: CG[15.87437616622; 7.97109594751]
Coordinates of the circumscribed circle: U[9.10987357682; 18.83439280906]
Coordinates of the inscribed circle: I[14.85987357682; 5.27992720615]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 64.93300086922° = 64°55'48″ = 2.00883502186 rad
∠ B' = β' = 140.88° = 140°52'48″ = 0.68327728034 rad
∠ C' = γ' = 154.1989991308° = 154°11'24″ = 0.45504696316 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 = 37.9 ; ; b = 26.4 ; ; c = 18.22 ; ;

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

p = a+b+c = 37.9+26.4+18.22 = 82.52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 82.52 }{ 2 } = 41.26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 41.26 * (41.26-37.9)(41.26-26.4)(41.26-18.22) } ; ; T = sqrt{ 47443.85 } = 217.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 217.82 }{ 37.9 } = 11.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 217.82 }{ 26.4 } = 16.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 217.82 }{ 18.22 } = 23.91 ; ;

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{ 37.9**2-26.4**2-18.22**2 }{ 2 * 26.4 * 18.22 } ) = 115° 4'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 26.4**2-37.9**2-18.22**2 }{ 2 * 37.9 * 18.22 } ) = 39° 7'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 18.22**2-37.9**2-26.4**2 }{ 2 * 26.4 * 37.9 } ) = 25° 48'36" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 217.82 }{ 41.26 } = 5.28 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37.9 }{ 2 * sin 115° 4'12" } = 20.92 ; ;





#2 Acute scalene triangle.

Sides: a = 37.9   b = 26.4   c = 40.59901553638

Area: T = 485.3143725243
Perimeter: p = 104.8990155364
Semiperimeter: s = 52.44550776819

Angle ∠ A = α = 64.93300086922° = 64°55'48″ = 1.1333242435 rad
Angle ∠ B = β = 39.12° = 39°7'12″ = 0.68327728034 rad
Angle ∠ C = γ = 75.95499913078° = 75°57' = 1.32655774152 rad

Height: ha = 25.61102229679
Height: hb = 36.76661913062
Height: hc = 23.91328784254

Median: ma = 28.51659228543
Median: mb = 36.98330414681
Median: mc = 25.5898958984

Inradius: r = 9.25437516711
Circumradius: R = 20.92109444007

Vertex coordinates: A[40.59901553638; 0] B[0; 0] C[29.40438134501; 23.91328784254]
Centroid: CG[23.3311322938; 7.97109594751]
Coordinates of the circumscribed circle: U[20.29550776819; 5.07989503348]
Coordinates of the inscribed circle: I[26.04550776819; 9.25437516711]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115.0769991308° = 115°4'12″ = 1.1333242435 rad
∠ B' = β' = 140.88° = 140°52'48″ = 0.68327728034 rad
∠ C' = γ' = 104.0550008692° = 104°3' = 1.32655774152 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 = 37.9 ; ; b = 26.4 ; ; c = 40.59 ; ;

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

p = a+b+c = 37.9+26.4+40.59 = 104.89 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 104.89 }{ 2 } = 52.45 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 52.45 * (52.45-37.9)(52.45-26.4)(52.45-40.59) } ; ; T = sqrt{ 235529.41 } = 485.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 485.31 }{ 37.9 } = 25.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 485.31 }{ 26.4 } = 36.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 485.31 }{ 40.59 } = 23.91 ; ;

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{ 37.9**2-26.4**2-40.59**2 }{ 2 * 26.4 * 40.59 } ) = 64° 55'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 26.4**2-37.9**2-40.59**2 }{ 2 * 37.9 * 40.59 } ) = 39° 7'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 40.59**2-37.9**2-26.4**2 }{ 2 * 26.4 * 37.9 } ) = 75° 57' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 485.31 }{ 52.45 } = 9.25 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37.9 }{ 2 * sin 64° 55'48" } = 20.92 ; ;




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