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=39.8; b=27.5; c=20.13876781907 and a=39.8; b=27.5; c=41.1076526391.

#1 Obtuse scalene triangle.

Sides: a = 39.8   b = 27.5   c = 20.13876781907

Area: T = 255.9879684879
Perimeter: p = 87.43876781907
Semiperimeter: s = 43.71988390953

Angle ∠ A = α = 112.4111242703° = 112°24'40″ = 1.96219463014 rad
Angle ∠ B = β = 39.7° = 39°42' = 0.6932895713 rad
Angle ∠ C = γ = 27.88987572966° = 27°53'20″ = 0.48767506391 rad

Height: ha = 12.86333007477
Height: hb = 18.61767043549
Height: hc = 25.42329591371

Median: ma = 13.59769864844
Median: mb = 28.38552169528
Median: mc = 32.6921948233

Inradius: r = 5.8555134541
Circumradius: R = 21.5265818338

Vertex coordinates: A[20.13876781907; 0] B[0; 0] C[30.62221022909; 25.42329591371]
Centroid: CG[16.92199268272; 8.47443197124]
Coordinates of the circumscribed circle: U[10.06988390953; 19.02657545026]
Coordinates of the inscribed circle: I[16.21988390953; 5.8555134541]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 67.58987572966° = 67°35'20″ = 1.96219463014 rad
∠ B' = β' = 140.3° = 140°18' = 0.6932895713 rad
∠ C' = γ' = 152.1111242703° = 152°6'40″ = 0.48767506391 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 = 39.8 ; ; b = 27.5 ; ; c = 20.14 ; ;

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

p = a+b+c = 39.8+27.5+20.14 = 87.44 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 87.44 }{ 2 } = 43.72 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 43.72 * (43.72-39.8)(43.72-27.5)(43.72-20.14) } ; ; T = sqrt{ 65525.6 } = 255.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 255.98 }{ 39.8 } = 12.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 255.98 }{ 27.5 } = 18.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 255.98 }{ 20.14 } = 25.42 ; ;

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{ 39.8**2-27.5**2-20.14**2 }{ 2 * 27.5 * 20.14 } ) = 112° 24'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27.5**2-39.8**2-20.14**2 }{ 2 * 39.8 * 20.14 } ) = 39° 42' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20.14**2-39.8**2-27.5**2 }{ 2 * 27.5 * 39.8 } ) = 27° 53'20" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 255.98 }{ 43.72 } = 5.86 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.8 }{ 2 * sin 112° 24'40" } = 21.53 ; ;





#2 Acute scalene triangle.

Sides: a = 39.8   b = 27.5   c = 41.1076526391

Area: T = 522.5254770354
Perimeter: p = 108.4076526391
Semiperimeter: s = 54.20332631955

Angle ∠ A = α = 67.58987572966° = 67°35'20″ = 1.18796463522 rad
Angle ∠ B = β = 39.7° = 39°42' = 0.6932895713 rad
Angle ∠ C = γ = 72.71112427034° = 72°42'40″ = 1.26990505884 rad

Height: ha = 26.25875261485
Height: hb = 38.00218014803
Height: hc = 25.42329591371

Median: ma = 28.75774034984
Median: mb = 38.05503712987
Median: mc = 27.3444256655

Inradius: r = 9.64400980227
Circumradius: R = 21.5265818338

Vertex coordinates: A[41.1076526391; 0] B[0; 0] C[30.62221022909; 25.42329591371]
Centroid: CG[23.9109542894; 8.47443197124]
Coordinates of the circumscribed circle: U[20.55332631955; 6.39772046345]
Coordinates of the inscribed circle: I[26.70332631955; 9.64400980227]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.4111242703° = 112°24'40″ = 1.18796463522 rad
∠ B' = β' = 140.3° = 140°18' = 0.6932895713 rad
∠ C' = γ' = 107.2898757297° = 107°17'20″ = 1.26990505884 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 = 39.8 ; ; b = 27.5 ; ; c = 41.11 ; ;

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

p = a+b+c = 39.8+27.5+41.11 = 108.41 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 108.41 }{ 2 } = 54.2 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 54.2 * (54.2-39.8)(54.2-27.5)(54.2-41.11) } ; ; T = sqrt{ 273032.14 } = 522.52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 522.52 }{ 39.8 } = 26.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 522.52 }{ 27.5 } = 38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 522.52 }{ 41.11 } = 25.42 ; ;

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{ 39.8**2-27.5**2-41.11**2 }{ 2 * 27.5 * 41.11 } ) = 67° 35'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27.5**2-39.8**2-41.11**2 }{ 2 * 39.8 * 41.11 } ) = 39° 42' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 41.11**2-39.8**2-27.5**2 }{ 2 * 27.5 * 39.8 } ) = 72° 42'40" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 522.52 }{ 54.2 } = 9.64 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.8 }{ 2 * sin 67° 35'20" } = 21.53 ; ;




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