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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 400**2 = 500**2 + c**2 -2 * 400 * c * cos (25° ) ; ; ; ; c**2 -906.308c +90000 =0 ; ; p=1; q=-906.307787037; 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} = 453.153893518 ± 339.629873849 ; ;
c_{1} = 792.783767368 ; ; c_{2} = 113.524019669 ; ; ; ; (c -792.783767368) (c -113.524019669) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 500**2-400**2-113.52**2 }{ 2 * 400 * 113.52 } ) = 148° 6'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 400**2-500**2-113.52**2 }{ 2 * 500 * 113.52 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 113.52**2-500**2-400**2 }{ 2 * 400 * 500 } ) = 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 ; ;




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