Triangle calculator SSA

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Triangle has two solutions with side c=133.012222075 and with side c=66.15993344611

#1 Acute scalene triangle.

Sides: a = 130   b = 90   c = 133.012222075

Area: T = 5557.409948327
Perimeter: p = 353.012222075
Semiperimeter: s = 176.5066110375

Angle ∠ A = α = 68.19877113074° = 68°11'52″ = 1.19902746046 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 71.80222886926° = 71°48'8″ = 1.25331863482 rad

Height: ha = 85.49986074349
Height: hb = 123.4987988517
Height: hc = 83.56223892593

Median: ma = 93.11988779701
Median: mb = 123.5766395134
Median: mc = 89.87217824614

Inradius: r = 31.48656492586
Circumradius: R = 70.00875722087

Vertex coordinates: A[133.012222075; 0] B[0; 0] C[99.58657776055; 83.56223892593]
Centroid: CG[77.53326661184; 27.85441297531]
Coordinates of the circumscribed circle: U[66.50661103749; 21.86331527771]
Coordinates of the inscribed circle: I[86.50661103749; 31.48656492586]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 111.8022288693° = 111°48'8″ = 1.19902746046 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 108.1987711307° = 108°11'52″ = 1.25331863482 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 130 ; ; b = 90 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 130**2 + c**2 -2 * 130 * c * cos (40° ) ; ; ; ; c**2 -199.172c +8800 =0 ; ; p=1; q=-199.172; r=8800 ; ; D = q**2 - 4pr = 199.172**2 - 4 * 1 * 8800 = 4469.30840514 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 199.17 ± sqrt{ 4469.31 } }{ 2 } ; ; c_{1,2} = 99.58577761 ± 33.4264431444 ; ; c_{1} = 133.012220754 ; ;
c_{2} = 66.1593344656 ; ; ; ; (c -133.012220754) (c -66.1593344656) = 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 = 130 ; ; b = 90 ; ; c = 133.01 ; ;

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

p = a+b+c = 130+90+133.01 = 353.01 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 353.01 }{ 2 } = 176.51 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 176.51 * (176.51-130)(176.51-90)(176.51-133.01) } ; ; T = sqrt{ 30884800.16 } = 5557.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5557.41 }{ 130 } = 85.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5557.41 }{ 90 } = 123.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5557.41 }{ 133.01 } = 83.56 ; ;

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{ 130**2-90**2-133.01**2 }{ 2 * 90 * 133.01 } ) = 68° 11'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-130**2-133.01**2 }{ 2 * 130 * 133.01 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 133.01**2-130**2-90**2 }{ 2 * 90 * 130 } ) = 71° 48'8" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5557.41 }{ 176.51 } = 31.49 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 130 }{ 2 * sin 68° 11'52" } = 70.01 ; ;





#2 Obtuse scalene triangle.

Sides: a = 130   b = 90   c = 66.15993344611

Area: T = 2764.216602968
Perimeter: p = 286.1599334461
Semiperimeter: s = 143.0879667231

Angle ∠ A = α = 111.8022288693° = 111°48'8″ = 1.9511318049 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 28.19877113074° = 28°11'52″ = 0.49221429038 rad

Height: ha = 42.52664004567
Height: hb = 61.42770228819
Height: hc = 83.56223892593

Median: ma = 44.8722360849
Median: mb = 92.80990985204
Median: mc = 106.7987638625

Inradius: r = 19.31994189167
Circumradius: R = 70.00875722087

Vertex coordinates: A[66.15993344611; 0] B[0; 0] C[99.58657776055; 83.56223892593]
Centroid: CG[55.24883706888; 27.85441297531]
Coordinates of the circumscribed circle: U[33.08796672305; 61.69992364821]
Coordinates of the inscribed circle: I[53.08796672305; 19.31994189167]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 68.19877113074° = 68°11'52″ = 1.9511318049 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 151.8022288693° = 151°48'8″ = 0.49221429038 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 130 ; ; b = 90 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 130**2 + c**2 -2 * 130 * c * cos (40° ) ; ; ; ; c**2 -199.172c +8800 =0 ; ; p=1; q=-199.172; r=8800 ; ; D = q**2 - 4pr = 199.172**2 - 4 * 1 * 8800 = 4469.30840514 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 199.17 ± sqrt{ 4469.31 } }{ 2 } ; ; c_{1,2} = 99.58577761 ± 33.4264431444 ; ; c_{1} = 133.012220754 ; ; : Nr. 1
c_{2} = 66.1593344656 ; ; ; ; (c -133.012220754) (c -66.1593344656) = 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 = 130 ; ; b = 90 ; ; c = 66.16 ; ;

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

p = a+b+c = 130+90+66.16 = 286.16 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 286.16 }{ 2 } = 143.08 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 143.08 * (143.08-130)(143.08-90)(143.08-66.16) } ; ; T = sqrt{ 7640890.26 } = 2764.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2764.22 }{ 130 } = 42.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2764.22 }{ 90 } = 61.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2764.22 }{ 66.16 } = 83.56 ; ;

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{ 130**2-90**2-66.16**2 }{ 2 * 90 * 66.16 } ) = 111° 48'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-130**2-66.16**2 }{ 2 * 130 * 66.16 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 66.16**2-130**2-90**2 }{ 2 * 90 * 130 } ) = 28° 11'52" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2764.22 }{ 143.08 } = 19.32 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 130 }{ 2 * sin 111° 48'8" } = 70.01 ; ;




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