Triangle calculator SSA

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Triangle has two solutions with side c=100.9733216448 and with side c=17.97550637234

#1 Obtuse scalene triangle.

Sides: a = 68   b = 53   c = 100.9733216448

Area: T = 1664.395474851
Perimeter: p = 221.9733216448
Semiperimeter: s = 110.9876608224

Angle ∠ A = α = 38.46437880528° = 38°27'50″ = 0.67113197443 rad
Angle ∠ B = β = 29° = 0.50661454831 rad
Angle ∠ C = γ = 112.5366211947° = 112°32'10″ = 1.96441274262 rad

Height: ha = 48.9532786721
Height: hb = 62.80773490005
Height: hc = 32.96770541768

Median: ma = 73.11883644503
Median: mb = 81.9899604516
Median: mc = 34.17701973957

Inradius: r = 14.99663565438
Circumradius: R = 54.66106315001

Vertex coordinates: A[100.9733216448; 0] B[0; 0] C[59.47441400855; 32.96770541768]
Centroid: CG[53.48224521777; 10.98990180589]
Coordinates of the circumscribed circle: U[50.48766082238; -20.95496306901]
Coordinates of the inscribed circle: I[57.98766082238; 14.99663565438]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.5366211947° = 141°32'10″ = 0.67113197443 rad
∠ B' = β' = 151° = 0.50661454831 rad
∠ C' = γ' = 67.46437880528° = 67°27'50″ = 1.96441274262 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 = 68 ; ; b = 53 ; ; c = 100.97 ; ;

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

p = a+b+c = 68+53+100.97 = 221.97 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 221.97 }{ 2 } = 110.99 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 110.99 * (110.99-68)(110.99-53)(110.99-100.97) } ; ; T = sqrt{ 2770209.88 } = 1664.39 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1664.39 }{ 68 } = 48.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1664.39 }{ 53 } = 62.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1664.39 }{ 100.97 } = 32.97 ; ;

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{ 68**2-53**2-100.97**2 }{ 2 * 53 * 100.97 } ) = 38° 27'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 53**2-68**2-100.97**2 }{ 2 * 68 * 100.97 } ) = 29° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 100.97**2-68**2-53**2 }{ 2 * 53 * 68 } ) = 112° 32'10" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1664.39 }{ 110.99 } = 15 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 68 }{ 2 * sin 38° 27'50" } = 54.66 ; ;





#2 Obtuse scalene triangle.

Sides: a = 68   b = 53   c = 17.97550637234

Area: T = 296.29224498
Perimeter: p = 138.9755063723
Semiperimeter: s = 69.48875318617

Angle ∠ A = α = 141.5366211947° = 141°32'10″ = 2.47702729093 rad
Angle ∠ B = β = 29° = 0.50661454831 rad
Angle ∠ C = γ = 9.46437880528° = 9°27'50″ = 0.16551742612 rad

Height: ha = 8.71444838176
Height: hb = 11.18108471623
Height: hc = 32.96770541768

Median: ma = 20.25497273545
Median: mb = 42.0876832358
Median: mc = 60.29769673453

Inradius: r = 4.26439656621
Circumradius: R = 54.66106315001

Vertex coordinates: A[17.97550637234; 0] B[0; 0] C[59.47441400855; 32.96770541768]
Centroid: CG[25.81664012696; 10.98990180589]
Coordinates of the circumscribed circle: U[8.98875318617; 53.91766848668]
Coordinates of the inscribed circle: I[16.48875318617; 4.26439656621]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 38.46437880528° = 38°27'50″ = 2.47702729093 rad
∠ B' = β' = 151° = 0.50661454831 rad
∠ C' = γ' = 170.5366211947° = 170°32'10″ = 0.16551742612 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 68 ; ; b = 53 ; ; beta = 29° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 53**2 = 68**2 + c**2 -2 * 53 * c * cos (29° ) ; ; ; ; c**2 -118.948c +1815 =0 ; ; p=1; q=-118.948280171; r=1815 ; ; D = q**2 - 4pr = 118.948**2 - 4 * 1 * 1815 = 6888.69335563 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 118.95 ± sqrt{ 6888.69 } }{ 2 } ; ; c_{1,2} = 59.4741400855 ± 41.4990763621 ; ; c_{1} = 100.973216448 ; ;
c_{2} = 17.9750637234 ; ; ; ; (c -100.973216448) (c -17.9750637234) = 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 = 68 ; ; b = 53 ; ; c = 17.98 ; ;

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

p = a+b+c = 68+53+17.98 = 138.98 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 138.98 }{ 2 } = 69.49 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 69.49 * (69.49-68)(69.49-53)(69.49-17.98) } ; ; T = sqrt{ 87789.22 } = 296.29 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 296.29 }{ 68 } = 8.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 296.29 }{ 53 } = 11.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 296.29 }{ 17.98 } = 32.97 ; ;

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{ 68**2-53**2-17.98**2 }{ 2 * 53 * 17.98 } ) = 141° 32'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 53**2-68**2-17.98**2 }{ 2 * 68 * 17.98 } ) = 29° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 17.98**2-68**2-53**2 }{ 2 * 53 * 68 } ) = 9° 27'50" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 296.29 }{ 69.49 } = 4.26 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 68 }{ 2 * sin 141° 32'10" } = 54.66 ; ;




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