Triangle calculator SSA

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Triangle has two solutions with side c=170.0911070979 and with side c=11.17704864286

#1 Obtuse scalene triangle.

Sides: a = 100   b = 90   c = 170.0911070979

Area: T = 3594.187963773
Perimeter: p = 360.0911070979
Semiperimeter: s = 180.0465535489

Angle ∠ A = α = 28.00767673296° = 28°24″ = 0.48988103027 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 126.993323267° = 126°59'36″ = 2.21664500378 rad

Height: ha = 71.88435927546
Height: hb = 79.87106586163
Height: hc = 42.26218261741

Median: ma = 126.5522306235
Median: mb = 132.0622433013
Median: mc = 42.62992961862

Inradius: r = 19.96326146128
Circumradius: R = 106.4799071242

Vertex coordinates: A[170.0911070979; 0] B[0; 0] C[90.63107787037; 42.26218261741]
Centroid: CG[86.90772832275; 14.08772753914]
Coordinates of the circumscribed circle: U[85.04655354893; -64.07106602577]
Coordinates of the inscribed circle: I[90.04655354893; 19.96326146128]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.993323267° = 151°59'36″ = 0.48988103027 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 53.00767673296° = 53°24″ = 2.21664500378 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 90 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 100**2 + c**2 -2 * 100 * c * cos (25° ) ; ; ; ; c**2 -181.262c +1900 =0 ; ; p=1; q=-181.262; r=1900 ; ; D = q**2 - 4pr = 181.262**2 - 4 * 1 * 1900 = 25255.7521937 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 181.26 ± sqrt{ 25255.75 } }{ 2 } ; ;
c_{1,2} = 90.6307787 ± 79.460292275 ; ; c_{1} = 170.091070979 ; ; c_{2} = 11.1704864286 ; ; ; ; text{ Factored form: } ; ; (c -170.091070979) (c -11.1704864286) = 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 = 100 ; ; b = 90 ; ; c = 170.09 ; ;

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

p = a+b+c = 100+90+170.09 = 360.09 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 360.09 }{ 2 } = 180.05 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 180.05 * (180.05-100)(180.05-90)(180.05-170.09) } ; ; T = sqrt{ 12918127.27 } = 3594.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3594.18 }{ 100 } = 71.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3594.18 }{ 90 } = 79.87 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3594.18 }{ 170.09 } = 42.26 ; ;

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+170.09**2-100**2 }{ 2 * 90 * 170.09 } ) = 28° 24" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+170.09**2-90**2 }{ 2 * 100 * 170.09 } ) = 25° ; ;
 gamma = 180° - alpha - beta = 180° - 28° 24" - 25° = 126° 59'36" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3594.18 }{ 180.05 } = 19.96 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 28° 24" } = 106.48 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 170.09**2 - 100**2 } }{ 2 } = 126.552 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 170.09**2+2 * 100**2 - 90**2 } }{ 2 } = 132.062 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 100**2 - 170.09**2 } }{ 2 } = 42.629 ; ;



#2 Obtuse scalene triangle.

Sides: a = 100   b = 90   c = 11.17704864286

Area: T = 236.0432577863
Perimeter: p = 201.1770486429
Semiperimeter: s = 100.5855243214

Angle ∠ A = α = 151.993323267° = 151°59'36″ = 2.65327823508 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 3.00767673296° = 3°24″ = 0.05224779897 rad

Height: ha = 4.72108515573
Height: hb = 5.24553906192
Height: hc = 42.26218261741

Median: ma = 40.15545748767
Median: mb = 55.11325202066
Median: mc = 94.96773894463

Inradius: r = 2.34766919234
Circumradius: R = 106.4799071242

Vertex coordinates: A[11.17704864286; 0] B[0; 0] C[90.63107787037; 42.26218261741]
Centroid: CG[33.93437550441; 14.08772753914]
Coordinates of the circumscribed circle: U[5.58552432143; 106.3322486432]
Coordinates of the inscribed circle: I[10.58552432143; 2.34766919234]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 28.00767673296° = 28°24″ = 2.65327823508 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 176.993323267° = 176°59'36″ = 0.05224779897 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 90 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 100**2 + c**2 -2 * 100 * c * cos (25° ) ; ; ; ; c**2 -181.262c +1900 =0 ; ; p=1; q=-181.262; r=1900 ; ; D = q**2 - 4pr = 181.262**2 - 4 * 1 * 1900 = 25255.7521937 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 181.26 ± sqrt{ 25255.75 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 90.6307787 ± 79.460292275 ; ; c_{1} = 170.091070979 ; ; c_{2} = 11.1704864286 ; ; ; ; text{ Factored form: } ; ; (c -170.091070979) (c -11.1704864286) = 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 = 100 ; ; b = 90 ; ; c = 11.17 ; ;

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

p = a+b+c = 100+90+11.17 = 201.17 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 201.17 }{ 2 } = 100.59 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 100.59 * (100.59-100)(100.59-90)(100.59-11.17) } ; ; T = sqrt{ 55716.1 } = 236.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 236.04 }{ 100 } = 4.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 236.04 }{ 90 } = 5.25 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 236.04 }{ 11.17 } = 42.26 ; ;

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+11.17**2-100**2 }{ 2 * 90 * 11.17 } ) = 151° 59'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+11.17**2-90**2 }{ 2 * 100 * 11.17 } ) = 25° ; ;
 gamma = 180° - alpha - beta = 180° - 151° 59'36" - 25° = 3° 24" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 236.04 }{ 100.59 } = 2.35 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 151° 59'36" } = 106.48 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 11.17**2 - 100**2 } }{ 2 } = 40.155 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.17**2+2 * 100**2 - 90**2 } }{ 2 } = 55.113 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 100**2 - 11.17**2 } }{ 2 } = 94.967 ; ;
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