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?

1. Use Law of Cosines

a = 110 ; ; b = 90 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 110**2 + c**2 -2 * 110 * c * cos (25° ) ; ; ; ; c**2 -199.388c +4000 =0 ; ; p=1; q=-199.388; 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} = 99.69385657 ± 77.0640320682 ; ; c_{1} = 176.757888638 ; ;
c_{2} = 22.6298245018 ; ; ; ; text{ Factored form: } ; ; (c -176.757888638) (c -22.6298245018) = 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 = 176.76 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 176.76**2 - 110**2 } }{ 2 } = 129.022 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 176.76**2+2 * 110**2 - 90**2 } }{ 2 } = 140.167 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 110**2 - 176.76**2 } }{ 2 } = 47.845 ; ;







#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 110 ; ; b = 90 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 110**2 + c**2 -2 * 110 * c * cos (25° ) ; ; ; ; c**2 -199.388c +4000 =0 ; ; p=1; q=-199.388; 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} = 99.69385657 ± 77.0640320682 ; ; c_{1} = 176.757888638 ; ; : Nr. 1
c_{2} = 22.6298245018 ; ; ; ; text{ Factored form: } ; ; (c -176.757888638) (c -22.6298245018) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 90**2+22.63**2-110**2 }{ 2 * 90 * 22.63 } ) = 148° 54' ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 110**2+22.63**2-90**2 }{ 2 * 110 * 22.63 } ) = 25° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 110**2+90**2-22.63**2 }{ 2 * 110 * 90 } ) = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 22.63**2 - 110**2 } }{ 2 } = 35.792 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.63**2+2 * 110**2 - 90**2 } }{ 2 } = 65.43 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 110**2 - 22.63**2 } }{ 2 } = 99.86 ; ;
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