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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 18.3**2 = 20.2**2 + c**2 -2 * 18.3 * c * cos (38° ) ; ; ; ; c**2 -31.836c +73.15 =0 ; ; p=1; q=-31.8356344457; 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} = 15.9178172229 ± 13.424861457 ; ;
c_{1} = 29.3426786799 ; ; c_{2} = 2.49295576583 ; ; ; ; (c -29.3426786799) (c -2.49295576583) = 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 = 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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