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?

1. Input data entered: side a, b and angle β.

a = 37.7 ; ; b = 32.9 ; ; beta = 31.2° ; ;

2. From angle β, side a and b we calculate c - by using the law of cosines and quadratic equation:

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 32.9**2 = 37.7**2 + c**2 - 2 * 37.7 * c * cos(31° 12') ; ; ; ; ; ; c**2 -64.494c +338.88 =0 ; ; a=1; b=-64.494; c=338.88 ; ; D = b**2 - 4ac = 64.494**2 - 4 * 1 * 338.88 = 2804.01604351 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 64.49 ± sqrt{ 2804.02 } }{ 2 } ; ; c_{1,2} = 32.24723261 ± 26.476480334 ; ; c_{1} = 58.723712944 ; ; c_{2} = 5.77075227598 ; ;
 ; ; (c -58.723712944) (c -5.77075227598) = 0 ; ; ; ; c > 0 ; ; ; ; c = 58.724 ; ;


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 ; ;

3. 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 ; ;

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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{ 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" ; ;

8. Inradius

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

9. 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?

1. Input data entered: side a, b and angle β.

a = 37.7 ; ; b = 32.9 ; ; beta = 31.2° ; ; : Nr. 1

2. From angle β, side a and b we calculate c - by using the law of cosines and quadratic equation:

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 32.9**2 = 37.7**2 + c**2 - 2 * 37.7 * c * cos(31° 12') ; ; ; ; ; ; c**2 -64.494c +338.88 =0 ; ; a=1; b=-64.494; c=338.88 ; ; D = b**2 - 4ac = 64.494**2 - 4 * 1 * 338.88 = 2804.01604351 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 64.49 ± sqrt{ 2804.02 } }{ 2 } ; ; c_{1,2} = 32.24723261 ± 26.476480334 ; ; c_{1} = 58.723712944 ; ; c_{2} = 5.77075227598 ; ; : Nr. 1
 ; ; (c -58.723712944) (c -5.77075227598) = 0 ; ; ; ; c > 0 ; ; ; ; c = 58.724 ; ; : Nr. 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 = 37.7 ; ; b = 32.9 ; ; c = 58.72 ; ;

3. 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 ; ;

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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{ 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" ; ;

8. Inradius

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

9. Circumradius

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




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