Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side b, c and angle β.

Triangle has two solutions: a=9.69549076429; b=64; c=71 and a=97.47438527487; b=64; c=71.

#1 Obtuse scalene triangle.

Sides: a = 9.69549076429   b = 64   c = 71

Area: T = 225.795532515
Perimeter: p = 144.6954907643
Semiperimeter: s = 72.34774538215

Angle ∠ A = α = 5.70435697408° = 5°42'13″ = 0.10995460711 rad
Angle ∠ B = β = 41° = 0.71655849933 rad
Angle ∠ C = γ = 133.2966430259° = 133°17'47″ = 2.32664615892 rad

Height: ha = 46.58801910583
Height: hb = 7.0566103911
Height: hc = 6.36604316944

Median: ma = 67.41766314158
Median: mb = 39.28773467812
Median: mc = 28.89219645767

Inradius: r = 3.12109850966
Circumradius: R = 48.77660987746

Vertex coordinates: A[71; 0] B[0; 0] C[7.31768396775; 6.36604316944]
Centroid: CG[26.10656132258; 2.12201438981]
Coordinates of the circumscribed circle: U[35.5; -33.44993320063]
Coordinates of the inscribed circle: I[8.34774538215; 3.12109850966]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 174.2966430259° = 174°17'47″ = 0.10995460711 rad
∠ B' = β' = 139° = 0.71655849933 rad
∠ C' = γ' = 46.70435697408° = 46°42'13″ = 2.32664615892 rad




How did we calculate this triangle?

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

b = 64 ; ; c = 71 ; ; beta = 41° ; ;

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 64**2 = 71**2 + a**2 - 2 * 71 * a * cos 41° ; ; ; ; ; ; a**2 -107.169a +945 =0 ; ; p=1; q=-107.169; r=945 ; ; D = q**2 - 4pr = 107.169**2 - 4 * 1 * 945 = 7705.14320388 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 107.17 ± sqrt{ 7705.14 } }{ 2 } ; ; a_{1,2} = 53.5843802 ± 43.8894725529 ; ; a_{1} = 97.4738527529 ; ; a_{2} = 9.69490764712 ; ; ; ;
 text{ Factored form: } ; ; (a -97.4738527529) (a -9.69490764712) = 0 ; ; ; ; a > 0 ; ; ; ; a = 97.474 ; ;
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 = 9.69 ; ; b = 64 ; ; c = 71 ; ;

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

p = a+b+c = 9.69+64+71 = 144.69 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 144.69 }{ 2 } = 72.35 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 72.35 * (72.35-9.69)(72.35-64)(72.35-71) } ; ; T = sqrt{ 50983.53 } = 225.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 225.8 }{ 9.69 } = 46.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 225.8 }{ 64 } = 7.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 225.8 }{ 71 } = 6.36 ; ;

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{ 64**2+71**2-9.69**2 }{ 2 * 64 * 71 } ) = 5° 42'13" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9.69**2+71**2-64**2 }{ 2 * 9.69 * 71 } ) = 41° ; ; gamma = 180° - alpha - beta = 180° - 5° 42'13" - 41° = 133° 17'47" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 225.8 }{ 72.35 } = 3.12 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 9.69 }{ 2 * sin 5° 42'13" } = 48.78 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 64**2+2 * 71**2 - 9.69**2 } }{ 2 } = 67.417 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 71**2+2 * 9.69**2 - 64**2 } }{ 2 } = 39.287 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 64**2+2 * 9.69**2 - 71**2 } }{ 2 } = 28.892 ; ;







#2 Obtuse scalene triangle.

Sides: a = 97.47438527487   b = 64   c = 71

Area: T = 2270.175534211
Perimeter: p = 232.4743852749
Semiperimeter: s = 116.2376926374

Angle ∠ A = α = 92.29664302592° = 92°17'47″ = 1.61108765959 rad
Angle ∠ B = β = 41° = 0.71655849933 rad
Angle ∠ C = γ = 46.70435697408° = 46°42'13″ = 0.81551310644 rad

Height: ha = 46.58801910583
Height: hb = 70.9432979441
Height: hc = 63.94986011863

Median: ma = 46.83217414536
Median: mb = 79.03884462451
Median: mc = 74.42199300244

Inradius: r = 19.53105864747
Circumradius: R = 48.77660987746

Vertex coordinates: A[71; 0] B[0; 0] C[73.56444504907; 63.94986011863]
Centroid: CG[48.18881501636; 21.31662003954]
Coordinates of the circumscribed circle: U[35.5; 33.44993320063]
Coordinates of the inscribed circle: I[52.23769263743; 19.53105864747]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 87.70435697408° = 87°42'13″ = 1.61108765959 rad
∠ B' = β' = 139° = 0.71655849933 rad
∠ C' = γ' = 133.2966430259° = 133°17'47″ = 0.81551310644 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 64 ; ; c = 71 ; ; beta = 41° ; ; : Nr. 1

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 64**2 = 71**2 + a**2 - 2 * 71 * a * cos 41° ; ; ; ; ; ; a**2 -107.169a +945 =0 ; ; p=1; q=-107.169; r=945 ; ; D = q**2 - 4pr = 107.169**2 - 4 * 1 * 945 = 7705.14320388 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 107.17 ± sqrt{ 7705.14 } }{ 2 } ; ; a_{1,2} = 53.5843802 ± 43.8894725529 ; ; a_{1} = 97.4738527529 ; ; a_{2} = 9.69490764712 ; ; ; ; : Nr. 1
 text{ Factored form: } ; ; (a -97.4738527529) (a -9.69490764712) = 0 ; ; ; ; a > 0 ; ; ; ; a = 97.474 ; ; : 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 = 97.47 ; ; b = 64 ; ; c = 71 ; ;

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

p = a+b+c = 97.47+64+71 = 232.47 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 232.47 }{ 2 } = 116.24 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 116.24 * (116.24-97.47)(116.24-64)(116.24-71) } ; ; T = sqrt{ 5153696.08 } = 2270.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2270.18 }{ 97.47 } = 46.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2270.18 }{ 64 } = 70.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2270.18 }{ 71 } = 63.95 ; ;

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{ 64**2+71**2-97.47**2 }{ 2 * 64 * 71 } ) = 92° 17'47" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 97.47**2+71**2-64**2 }{ 2 * 97.47 * 71 } ) = 41° ; ; gamma = 180° - alpha - beta = 180° - 92° 17'47" - 41° = 46° 42'13" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2270.18 }{ 116.24 } = 19.53 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 97.47 }{ 2 * sin 92° 17'47" } = 48.78 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 64**2+2 * 71**2 - 97.47**2 } }{ 2 } = 46.832 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 71**2+2 * 97.47**2 - 64**2 } }{ 2 } = 79.038 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 64**2+2 * 97.47**2 - 71**2 } }{ 2 } = 74.42 ; ;
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