Triangle calculator SSA

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Triangle has two solutions with side c=792.7843767367 and with side c=113.5244019669

#1 Obtuse scalene triangle.

Sides: a = 500   b = 400   c = 792.7843767367

Area: T = 83761.22444253
Perimeter: p = 1692.784376737
Semiperimeter: s = 846.3921883684

Angle ∠ A = α = 31.88988308305° = 31°53'20″ = 0.55765650926 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 123.1111169169° = 123°6'40″ = 2.1498695248 rad

Height: ha = 335.0454897701
Height: hb = 418.8066122126
Height: hc = 211.309913087

Median: ma = 575.9880078562
Median: mb = 631.8654740986
Median: mc = 218.8800078953

Inradius: r = 98.96326980598
Circumradius: R = 473.244031663

Vertex coordinates: A[792.7843767367; 0] B[0; 0] C[453.1543893518; 211.309913087]
Centroid: CG[415.3132553629; 70.43663769568]
Coordinates of the circumscribed circle: U[396.3921883684; -258.5154742006]
Coordinates of the inscribed circle: I[446.3921883684; 98.96326980598]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.1111169169° = 148°6'40″ = 0.55765650926 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 56.88988308305° = 56°53'20″ = 2.1498695248 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 500 ; ; b = 400 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 400**2 = 500**2 + c**2 -2 * 500 * c * cos (25° ) ; ; ; ; c**2 -906.308c +90000 =0 ; ; p=1; q=-906.308; r=90000 ; ; D = q**2 - 4pr = 906.308**2 - 4 * 1 * 90000 = 461393.804843 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 906.31 ± sqrt{ 461393.8 } }{ 2 } ; ; c_{1,2} = 453.15389352 ± 339.629873849 ; ; c_{1} = 792.783767369 ; ;
c_{2} = 113.524019671 ; ; ; ; text{ Factored form: } ; ; (c -792.783767369) (c -113.524019671) = 0 ; ; ; ; c>0 ; ;
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 = 500 ; ; b = 400 ; ; c = 792.78 ; ;

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

p = a+b+c = 500+400+792.78 = 1692.78 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1692.78 }{ 2 } = 846.39 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 846.39 * (846.39-500)(846.39-400)(846.39-792.78) } ; ; T = sqrt{ 7015942717.22 } = 83761.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 83761.22 }{ 500 } = 335.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 83761.22 }{ 400 } = 418.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 83761.22 }{ 792.78 } = 211.31 ; ;

6. 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{ 400**2+792.78**2-500**2 }{ 2 * 400 * 792.78 } ) = 31° 53'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 500**2+792.78**2-400**2 }{ 2 * 500 * 792.78 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 31° 53'20" - 25° = 123° 6'40" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 83761.22 }{ 846.39 } = 98.96 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 500 }{ 2 * sin 31° 53'20" } = 473.24 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 400**2+2 * 792.78**2 - 500**2 } }{ 2 } = 575.98 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 792.78**2+2 * 500**2 - 400**2 } }{ 2 } = 631.865 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 400**2+2 * 500**2 - 792.78**2 } }{ 2 } = 218.8 ; ;





#2 Obtuse scalene triangle.

Sides: a = 500   b = 400   c = 113.5244019669

Area: T = 11994.33109646
Perimeter: p = 1013.524401967
Semiperimeter: s = 506.7622009835

Angle ∠ A = α = 148.1111169169° = 148°6'40″ = 2.5855027561 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 6.88988308305° = 6°53'20″ = 0.12202327796 rad

Height: ha = 47.97773238584
Height: hb = 59.9721654823
Height: hc = 211.309913087

Median: ma = 154.7388009296
Median: mb = 302.3976844429
Median: mc = 449.1977144069

Inradius: r = 23.66985677534
Circumradius: R = 473.244031663

Vertex coordinates: A[113.5244019669; 0] B[0; 0] C[453.1543893518; 211.309913087]
Centroid: CG[188.8932637729; 70.43663769568]
Coordinates of the circumscribed circle: U[56.76220098346; 469.8243872876]
Coordinates of the inscribed circle: I[106.7622009835; 23.66985677534]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 31.88988308305° = 31°53'20″ = 2.5855027561 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 173.1111169169° = 173°6'40″ = 0.12202327796 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 500 ; ; b = 400 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 400**2 = 500**2 + c**2 -2 * 500 * c * cos (25° ) ; ; ; ; c**2 -906.308c +90000 =0 ; ; p=1; q=-906.308; r=90000 ; ; D = q**2 - 4pr = 906.308**2 - 4 * 1 * 90000 = 461393.804843 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 906.31 ± sqrt{ 461393.8 } }{ 2 } ; ; c_{1,2} = 453.15389352 ± 339.629873849 ; ; c_{1} = 792.783767369 ; ; : Nr. 1
c_{2} = 113.524019671 ; ; ; ; text{ Factored form: } ; ; (c -792.783767369) (c -113.524019671) = 0 ; ; ; ; c>0 ; ; : 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 = 500 ; ; b = 400 ; ; c = 113.52 ; ;

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

p = a+b+c = 500+400+113.52 = 1013.52 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1013.52 }{ 2 } = 506.76 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 506.76 * (506.76-500)(506.76-400)(506.76-113.52) } ; ; T = sqrt{ 143863975.29 } = 11994.33 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11994.33 }{ 500 } = 47.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11994.33 }{ 400 } = 59.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11994.33 }{ 113.52 } = 211.31 ; ;

6. 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{ 400**2+113.52**2-500**2 }{ 2 * 400 * 113.52 } ) = 148° 6'40" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 500**2+113.52**2-400**2 }{ 2 * 500 * 113.52 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 148° 6'40" - 25° = 6° 53'20" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11994.33 }{ 506.76 } = 23.67 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 500 }{ 2 * sin 148° 6'40" } = 473.24 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 400**2+2 * 113.52**2 - 500**2 } }{ 2 } = 154.738 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 113.52**2+2 * 500**2 - 400**2 } }{ 2 } = 302.397 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 400**2+2 * 500**2 - 113.52**2 } }{ 2 } = 449.197 ; ;
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