Triangle calculator SSA

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Triangle has two solutions with side c=176.7587888642 and with side c=22.63298245059

#1 Obtuse scalene triangle.

Sides: a = 110   b = 90   c = 176.7587888642

Area: T = 4108.561114058
Perimeter: p = 376.7587888642
Semiperimeter: s = 188.3798944321

Angle ∠ A = α = 31.11000068352° = 31°6' = 0.54327975167 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 123.9899993165° = 123°54' = 2.16224628239 rad

Height: ha = 74.70111116469
Height: hb = 91.30113586796
Height: hc = 46.48880087915

Median: ma = 129.0221996569
Median: mb = 140.167659944
Median: mc = 47.84551899431

Inradius: r = 21.81100868724
Circumradius: R = 106.4799071242

Vertex coordinates: A[176.7587888642; 0] B[0; 0] C[99.6943856574; 46.48880087915]
Centroid: CG[92.15105817387; 15.49660029305]
Coordinates of the circumscribed circle: U[88.37989443211; -59.38881706505]
Coordinates of the inscribed circle: I[98.37989443211; 21.81100868724]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.9899993165° = 148°54' = 0.54327975167 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 56.11000068352° = 56°6' = 2.16224628239 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 = 110 ; ; b = 90 ; ; c = 176.76 ; ;

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

p = a+b+c = 110+90+176.76 = 376.76 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 376.76 }{ 2 } = 188.38 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 188.38 * (188.38-110)(188.38-90)(188.38-176.76) } ; ; T = sqrt{ 16880274.65 } = 4108.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4108.56 }{ 110 } = 74.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4108.56 }{ 90 } = 91.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4108.56 }{ 176.76 } = 46.49 ; ;

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{ 110**2-90**2-176.76**2 }{ 2 * 90 * 176.76 } ) = 31° 6' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-110**2-176.76**2 }{ 2 * 110 * 176.76 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 176.76**2-110**2-90**2 }{ 2 * 90 * 110 } ) = 123° 54' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4108.56 }{ 188.38 } = 21.81 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 110 }{ 2 * sin 31° 6' } = 106.48 ; ;





#2 Obtuse scalene triangle.

Sides: a = 110   b = 90   c = 22.63298245059

Area: T = 526.0087740289
Perimeter: p = 222.6329824506
Semiperimeter: s = 111.3154912253

Angle ∠ A = α = 148.9899993165° = 148°54' = 2.59987951369 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 6.11000068352° = 6°6' = 0.10664652037 rad

Height: ha = 9.56437770962
Height: hb = 11.68990608953
Height: hc = 46.48880087915

Median: ma = 35.79218213924
Median: mb = 65.43297675266
Median: mc = 99.86597654749

Inradius: r = 4.72554022812
Circumradius: R = 106.4799071242

Vertex coordinates: A[22.63298245059; 0] B[0; 0] C[99.6943856574; 46.48880087915]
Centroid: CG[40.775456036; 15.49660029305]
Coordinates of the circumscribed circle: U[11.31549122529; 105.8766179442]
Coordinates of the inscribed circle: I[21.31549122529; 4.72554022812]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 31.11000068352° = 31°6' = 2.59987951369 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 173.9899993165° = 173°54' = 0.10664652037 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 110 ; ; b = 90 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 110**2 + c**2 -2 * 90 * c * cos (25° ) ; ; ; ; c**2 -199.388c +4000 =0 ; ; p=1; q=-199.387713148; r=4000 ; ; D = q**2 - 4pr = 199.388**2 - 4 * 1 * 4000 = 23755.4601544 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 199.39 ± sqrt{ 23755.46 } }{ 2 } ; ; c_{1,2} = 99.693856574 ± 77.0640320682 ; ; c_{1} = 176.757888642 ; ;
c_{2} = 22.6298245059 ; ; ; ; (c -176.757888642) (c -22.6298245059) = 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 = 110 ; ; b = 90 ; ; c = 22.63 ; ;

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

p = a+b+c = 110+90+22.63 = 222.63 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 222.63 }{ 2 } = 111.31 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 111.31 * (111.31-110)(111.31-90)(111.31-22.63) } ; ; T = sqrt{ 276684.14 } = 526.01 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 526.01 }{ 110 } = 9.56 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 526.01 }{ 90 } = 11.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 526.01 }{ 22.63 } = 46.49 ; ;

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{ 110**2-90**2-22.63**2 }{ 2 * 90 * 22.63 } ) = 148° 54' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-110**2-22.63**2 }{ 2 * 110 * 22.63 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22.63**2-110**2-90**2 }{ 2 * 90 * 110 } ) = 6° 6' ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 526.01 }{ 111.31 } = 4.73 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 110 }{ 2 * sin 148° 54' } = 106.48 ; ;




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