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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 90**2 = 130**2 + c**2 -2 * 90 * c * cos (40° ) ; ; ; ; c**2 -199.172c +8800 =0 ; ; p=1; q=-199.171555211; 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} = 99.5857776055 ± 33.4264431444 ; ; c_{1} = 133.01222075 ; ;
c_{2} = 66.1593344611 ; ; ; ; (c -133.01222075) (c -66.1593344611) = 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 = 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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