Triangle calculator SSA

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Triangle has two solutions with side c=29.34326786799 and with side c=2.49329557658

#1 Obtuse scalene triangle.

Sides: a = 20.2   b = 18.3   c = 29.34326786799

Area: T = 182.4588084145
Perimeter: p = 67.84326786799
Semiperimeter: s = 33.92113393399

Angle ∠ A = α = 42.81110354856° = 42°48'40″ = 0.74771935254 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 99.18989645144° = 99°11'20″ = 1.73111740124 rad

Height: ha = 18.06551568461
Height: hb = 19.94107742235
Height: hc = 12.43663618016

Median: ma = 22.2769517194
Median: mb = 23.46989986164
Median: mc = 12.49986720084

Inradius: r = 5.37988584913
Circumradius: R = 14.86220635962

Vertex coordinates: A[29.34326786799; 0] B[0; 0] C[15.91878172229; 12.43663618016]
Centroid: CG[15.08768319676; 4.14554539339]
Coordinates of the circumscribed circle: U[14.67113393399; -2.37333386418]
Coordinates of the inscribed circle: I[15.62113393399; 5.37988584913]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.1898964514° = 137°11'20″ = 0.74771935254 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 80.81110354856° = 80°48'40″ = 1.73111740124 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 20.2 ; ; b = 18.3 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 18.3**2 = 20.2**2 + c**2 -2 * 20.2 * c * cos (38° ) ; ; ; ; c**2 -31.836c +73.15 =0 ; ; p=1; q=-31.836; r=73.15 ; ; D = q**2 - 4pr = 31.836**2 - 4 * 1 * 73.15 = 720.907620561 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 31.84 ± sqrt{ 720.91 } }{ 2 } ; ; c_{1,2} = 15.91781722 ± 13.424861457 ; ; c_{1} = 29.342678677 ; ;
c_{2} = 2.49295576298 ; ; ; ; (c -29.342678677) (c -2.49295576298) = 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 = 20.2 ; ; b = 18.3 ; ; c = 29.34 ; ;

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

p = a+b+c = 20.2+18.3+29.34 = 67.84 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67.84 }{ 2 } = 33.92 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.92 * (33.92-20.2)(33.92-18.3)(33.92-29.34) } ; ; T = sqrt{ 33290.95 } = 182.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 182.46 }{ 20.2 } = 18.07 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 182.46 }{ 18.3 } = 19.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 182.46 }{ 29.34 } = 12.44 ; ;

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{ 20.2**2-18.3**2-29.34**2 }{ 2 * 18.3 * 29.34 } ) = 42° 48'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18.3**2-20.2**2-29.34**2 }{ 2 * 20.2 * 29.34 } ) = 38° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29.34**2-20.2**2-18.3**2 }{ 2 * 18.3 * 20.2 } ) = 99° 11'20" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 182.46 }{ 33.92 } = 5.38 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20.2 }{ 2 * sin 42° 48'40" } = 14.86 ; ;





#2 Obtuse scalene triangle.

Sides: a = 20.2   b = 18.3   c = 2.49329557658

Area: T = 15.50216499296
Perimeter: p = 40.99329557658
Semiperimeter: s = 20.49664778829

Angle ∠ A = α = 137.1898964514° = 137°11'20″ = 2.39443991282 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 4.81110354856° = 4°48'40″ = 0.08439684097 rad

Height: ha = 1.53548168247
Height: hb = 1.69441693912
Height: hc = 12.43663618016

Median: ma = 8.27990346192
Median: mb = 11.10987764504
Median: mc = 19.23330780919

Inradius: r = 0.75663079871
Circumradius: R = 14.86220635962

Vertex coordinates: A[2.49329557658; 0] B[0; 0] C[15.91878172229; 12.43663618016]
Centroid: CG[6.13769243296; 4.14554539339]
Coordinates of the circumscribed circle: U[1.24664778829; 14.81097004434]
Coordinates of the inscribed circle: I[2.19664778829; 0.75663079871]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 42.81110354856° = 42°48'40″ = 2.39443991282 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 175.1898964514° = 175°11'20″ = 0.08439684097 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 20.2 ; ; b = 18.3 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 18.3**2 = 20.2**2 + c**2 -2 * 20.2 * c * cos (38° ) ; ; ; ; c**2 -31.836c +73.15 =0 ; ; p=1; q=-31.836; r=73.15 ; ; D = q**2 - 4pr = 31.836**2 - 4 * 1 * 73.15 = 720.907620561 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 31.84 ± sqrt{ 720.91 } }{ 2 } ; ; c_{1,2} = 15.91781722 ± 13.424861457 ; ; c_{1} = 29.342678677 ; ; : Nr. 1
c_{2} = 2.49295576298 ; ; ; ; (c -29.342678677) (c -2.49295576298) = 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 = 20.2 ; ; b = 18.3 ; ; c = 2.49 ; ;

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

p = a+b+c = 20.2+18.3+2.49 = 40.99 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40.99 }{ 2 } = 20.5 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.5 * (20.5-20.2)(20.5-18.3)(20.5-2.49) } ; ; T = sqrt{ 240.3 } = 15.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 15.5 }{ 20.2 } = 1.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 15.5 }{ 18.3 } = 1.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 15.5 }{ 2.49 } = 12.44 ; ;

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{ 20.2**2-18.3**2-2.49**2 }{ 2 * 18.3 * 2.49 } ) = 137° 11'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18.3**2-20.2**2-2.49**2 }{ 2 * 20.2 * 2.49 } ) = 38° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.49**2-20.2**2-18.3**2 }{ 2 * 18.3 * 20.2 } ) = 4° 48'40" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 15.5 }{ 20.5 } = 0.76 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20.2 }{ 2 * sin 137° 11'20" } = 14.86 ; ;




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