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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 90 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 100**2 + c**2 -2 * 90 * c * cos (25° ) ; ; ; ; c**2 -181.262c +1900 =0 ; ; p=1; q=-181.261557407; 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} = 90.6307787037 ± 79.460292275 ; ; c_{1} = 170.091070979 ; ;
c_{2} = 11.1704864286 ; ; ; ; (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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 100**2-90**2-11.17**2 }{ 2 * 90 * 11.17 } ) = 151° 59'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-100**2-11.17**2 }{ 2 * 100 * 11.17 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.17**2-100**2-90**2 }{ 2 * 90 * 100 } ) = 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 ; ;




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