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=73; b=63; c=11.23113542608 and a=73; b=63; c=121.0989582647.

#1 Obtuse scalene triangle.

Sides: a = 73   b = 63   c = 11.23113542608

Area: T = 173.2550002637
Perimeter: p = 147.2311354261
Semiperimeter: s = 73.61656771304

Angle ∠ A = α = 150.6799100112° = 150°40'45″ = 2.63298464109 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 4.32108998877° = 4°19'15″ = 0.07554139297 rad

Height: ha = 4.74765754147
Height: hb = 5.55000000837
Height: hc = 30.85111331071

Median: ma = 26.74554979252
Median: mb = 41.65771921673
Median: mc = 67.95219254353

Inradius: r = 2.35334389602
Circumradius: R = 74.53553498693

Vertex coordinates: A[11.23113542608; 0] B[0; 0] C[66.16604684537; 30.85111331071]
Centroid: CG[25.79772742382; 10.28437110357]
Coordinates of the circumscribed circle: U[5.61656771304; 74.32334993155]
Coordinates of the inscribed circle: I[10.61656771304; 2.35334389602]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.32108998877° = 29°19'15″ = 2.63298464109 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 175.6799100112° = 175°40'45″ = 0.07554139297 rad




How did we calculate this triangle?

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

a = 73 ; ; b = 63 ; ; beta = 25° ; ;

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 ; ; ; ; 63**2 = 73**2 + c**2 - 2 * 73 * c * cos(25° ) ; ; ; ; ; ; c**2 -132.321c +1360 =0 ; ; a=1; b=-132.321; c=1360 ; ; D = b**2 - 4ac = 132.321**2 - 4 * 1 * 1360 = 12068.830344 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 132.32 ± sqrt{ 12068.83 } }{ 2 } ; ; c_{1,2} = 66.16046845 ± 54.9291141928 ; ; c_{1} = 121.089582643 ; ; c_{2} = 11.2313542572 ; ;
 ; ; text{ Factored form: } ; ; (c -121.089582643) (c -11.2313542572) = 0 ; ; ; ; c > 0 ; ; ; ; c = 121.09 ; ;


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 = 73 ; ; b = 63 ; ; c = 11.23 ; ;

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

p = a+b+c = 73+63+11.23 = 147.23 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 147.23 }{ 2 } = 73.62 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 73.62 * (73.62-73)(73.62-63)(73.62-11.23) } ; ; T = sqrt{ 30015.56 } = 173.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 173.25 }{ 73 } = 4.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 173.25 }{ 63 } = 5.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 173.25 }{ 11.23 } = 30.85 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 63**2+11.23**2-73**2 }{ 2 * 63 * 11.23 } ) = 150° 40'45" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 73**2+11.23**2-63**2 }{ 2 * 73 * 11.23 } ) = 25° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 73**2+63**2-11.23**2 }{ 2 * 73 * 63 } ) = 4° 19'15" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 173.25 }{ 73.62 } = 2.35 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 73 }{ 2 * sin 150° 40'45" } = 74.54 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 63**2+2 * 11.23**2 - 73**2 } }{ 2 } = 26.745 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.23**2+2 * 73**2 - 63**2 } }{ 2 } = 41.657 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 63**2+2 * 73**2 - 11.23**2 } }{ 2 } = 67.952 ; ;







#2 Obtuse scalene triangle.

Sides: a = 73   b = 63   c = 121.0989582647

Area: T = 1867.875541605
Perimeter: p = 257.0989582646
Semiperimeter: s = 128.5454791323

Angle ∠ A = α = 29.32108998877° = 29°19'15″ = 0.51217462427 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 125.6799100112° = 125°40'45″ = 2.19435140979 rad

Height: ha = 51.1754668933
Height: hb = 59.29876322557
Height: hc = 30.85111331071

Median: ma = 89.35109569773
Median: mb = 94.88772673901
Median: mc = 31.35880650491

Inradius: r = 14.53109304004
Circumradius: R = 74.53553498693

Vertex coordinates: A[121.0989582647; 0] B[0; 0] C[66.16604684537; 30.85111331071]
Centroid: CG[62.41766837001; 10.28437110357]
Coordinates of the circumscribed circle: U[60.54547913233; -43.47223662085]
Coordinates of the inscribed circle: I[65.54547913233; 14.53109304004]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.6799100112° = 150°40'45″ = 0.51217462427 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 54.32108998877° = 54°19'15″ = 2.19435140979 rad

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How did we calculate this triangle?

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

a = 73 ; ; b = 63 ; ; beta = 25° ; ; : 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 ; ; ; ; 63**2 = 73**2 + c**2 - 2 * 73 * c * cos(25° ) ; ; ; ; ; ; c**2 -132.321c +1360 =0 ; ; a=1; b=-132.321; c=1360 ; ; D = b**2 - 4ac = 132.321**2 - 4 * 1 * 1360 = 12068.830344 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 132.32 ± sqrt{ 12068.83 } }{ 2 } ; ; c_{1,2} = 66.16046845 ± 54.9291141928 ; ; c_{1} = 121.089582643 ; ; c_{2} = 11.2313542572 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (c -121.089582643) (c -11.2313542572) = 0 ; ; ; ; c > 0 ; ; ; ; c = 121.09 ; ; : 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 = 73 ; ; b = 63 ; ; c = 121.09 ; ;

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

p = a+b+c = 73+63+121.09 = 257.09 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 257.09 }{ 2 } = 128.54 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 128.54 * (128.54-73)(128.54-63)(128.54-121.09) } ; ; T = sqrt{ 3488958.57 } = 1867.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1867.88 }{ 73 } = 51.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1867.88 }{ 63 } = 59.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1867.88 }{ 121.09 } = 30.85 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 63**2+121.09**2-73**2 }{ 2 * 63 * 121.09 } ) = 29° 19'15" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 73**2+121.09**2-63**2 }{ 2 * 73 * 121.09 } ) = 25° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 73**2+63**2-121.09**2 }{ 2 * 73 * 63 } ) = 125° 40'45" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1867.88 }{ 128.54 } = 14.53 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 73 }{ 2 * sin 29° 19'15" } = 74.54 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 63**2+2 * 121.09**2 - 73**2 } }{ 2 } = 89.351 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 121.09**2+2 * 73**2 - 63**2 } }{ 2 } = 94.887 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 63**2+2 * 73**2 - 121.09**2 } }{ 2 } = 31.358 ; ;
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