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=37.7; b=32.9; c=5.7710752274 and a=37.7; b=32.9; c=58.72437129421.

#1 Obtuse scalene triangle.

Sides: a = 37.7   b = 32.9   c = 5.7710752274

Area: T = 56.35502944733
Perimeter: p = 76.3710752274
Semiperimeter: s = 38.1855376137

Angle ∠ A = α = 143.5876720761° = 143°35'12″ = 2.50660610394 rad
Angle ∠ B = β = 31.2° = 31°12' = 0.54545427266 rad
Angle ∠ C = γ = 5.21332792389° = 5°12'48″ = 0.09109888875 rad

Height: ha = 2.98994055423
Height: hb = 3.4265549816
Height: hc = 19.53296182534

Median: ma = 14.23114191458
Median: mb = 21.37703834992
Median: mc = 35.26436442324

Inradius: r = 1.4765703533
Circumradius: R = 31.75551009935

Vertex coordinates: A[5.7710752274; 0] B[0; 0] C[32.24772326081; 19.53296182534]
Centroid: CG[12.67326616274; 6.51098727511]
Coordinates of the circumscribed circle: U[2.8855376137; 31.62437417719]
Coordinates of the inscribed circle: I[5.2855376137; 1.4765703533]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 36.41332792389° = 36°24'48″ = 2.50660610394 rad
∠ B' = β' = 148.8° = 148°48' = 0.54545427266 rad
∠ C' = γ' = 174.7876720761° = 174°47'12″ = 0.09109888875 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 = 37.7 ; ; b = 32.9 ; ; c = 5.77 ; ;

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

p = a+b+c = 37.7+32.9+5.77 = 76.37 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76.37 }{ 2 } = 38.19 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.19 * (38.19-37.7)(38.19-32.9)(38.19-5.77) } ; ; T = sqrt{ 3175.36 } = 56.35 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 56.35 }{ 37.7 } = 2.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 56.35 }{ 32.9 } = 3.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 56.35 }{ 5.77 } = 19.53 ; ;

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{ 37.7**2-32.9**2-5.77**2 }{ 2 * 32.9 * 5.77 } ) = 143° 35'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32.9**2-37.7**2-5.77**2 }{ 2 * 37.7 * 5.77 } ) = 31° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.77**2-37.7**2-32.9**2 }{ 2 * 32.9 * 37.7 } ) = 5° 12'48" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 56.35 }{ 38.19 } = 1.48 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37.7 }{ 2 * sin 143° 35'12" } = 31.76 ; ;





#2 Obtuse scalene triangle.

Sides: a = 37.7   b = 32.9   c = 58.72437129421

Area: T = 573.426584809
Perimeter: p = 129.3243712942
Semiperimeter: s = 64.6621856471

Angle ∠ A = α = 36.41332792389° = 36°24'48″ = 0.63655316142 rad
Angle ∠ B = β = 31.2° = 31°12' = 0.54545427266 rad
Angle ∠ C = γ = 112.3876720761° = 112°23'12″ = 1.96215183128 rad

Height: ha = 30.42204693947
Height: hb = 34.85987141696
Height: hc = 19.53296182534

Median: ma = 43.70549165524
Median: mb = 46.52218199435
Median: mc = 19.74216155513

Inradius: r = 8.86880696687
Circumradius: R = 31.75551009935

Vertex coordinates: A[58.72437129421; 0] B[0; 0] C[32.24772326081; 19.53296182534]
Centroid: CG[30.32436485167; 6.51098727511]
Coordinates of the circumscribed circle: U[29.3621856471; -12.09441235185]
Coordinates of the inscribed circle: I[31.7621856471; 8.86880696687]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.5876720761° = 143°35'12″ = 0.63655316142 rad
∠ B' = β' = 148.8° = 148°48' = 0.54545427266 rad
∠ C' = γ' = 67.61332792389° = 67°36'48″ = 1.96215183128 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 = 37.7 ; ; b = 32.9 ; ; c = 58.72 ; ;

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

p = a+b+c = 37.7+32.9+58.72 = 129.32 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 129.32 }{ 2 } = 64.66 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 64.66 * (64.66-37.7)(64.66-32.9)(64.66-58.72) } ; ; T = sqrt{ 328817.2 } = 573.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 573.43 }{ 37.7 } = 30.42 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 573.43 }{ 32.9 } = 34.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 573.43 }{ 58.72 } = 19.53 ; ;

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{ 37.7**2-32.9**2-58.72**2 }{ 2 * 32.9 * 58.72 } ) = 36° 24'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32.9**2-37.7**2-58.72**2 }{ 2 * 37.7 * 58.72 } ) = 31° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 58.72**2-37.7**2-32.9**2 }{ 2 * 32.9 * 37.7 } ) = 112° 23'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 573.43 }{ 64.66 } = 8.87 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37.7 }{ 2 * sin 36° 24'48" } = 31.76 ; ;




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