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?

1. Use Law of Cosines

a = 68 ; ; b = 53 ; ; beta = 29° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 53**2 = 68**2 + c**2 -2 * 68 * c * cos (29° ) ; ; ; ; c**2 -118.948c +1815 =0 ; ; p=1; q=-118.948; 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} = 59.47414009 ± 41.4990763621 ; ; c_{1} = 100.973216452 ; ; c_{2} = 17.9750637279 ; ; ; ; text{ Factored form: } ; ; (c -100.973216452) (c -17.9750637279) = 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 = 100.97 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 53**2+100.97**2-68**2 }{ 2 * 53 * 100.97 } ) = 38° 27'50" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 68**2+100.97**2-53**2 }{ 2 * 68 * 100.97 } ) = 29° ; ; gamma = 180° - alpha - beta = 180° - 38° 27'50" - 29° = 112° 32'10" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 53**2+2 * 100.97**2 - 68**2 } }{ 2 } = 73.118 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 100.97**2+2 * 68**2 - 53**2 } }{ 2 } = 81.9 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 53**2+2 * 68**2 - 100.97**2 } }{ 2 } = 34.17 ; ;







#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 68 ; ; b = 53 ; ; beta = 29° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 53**2 = 68**2 + c**2 -2 * 68 * c * cos (29° ) ; ; ; ; c**2 -118.948c +1815 =0 ; ; p=1; q=-118.948; 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} = 59.47414009 ± 41.4990763621 ; ; c_{1} = 100.973216452 ; ; c_{2} = 17.9750637279 ; ; ; ; text{ Factored form: } ; ; (c -100.973216452) (c -17.9750637279) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 53**2+17.98**2-68**2 }{ 2 * 53 * 17.98 } ) = 141° 32'10" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 68**2+17.98**2-53**2 }{ 2 * 68 * 17.98 } ) = 29° ; ; gamma = 180° - alpha - beta = 180° - 141° 32'10" - 29° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 53**2+2 * 17.98**2 - 68**2 } }{ 2 } = 20.25 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.98**2+2 * 68**2 - 53**2 } }{ 2 } = 42.087 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 53**2+2 * 68**2 - 17.98**2 } }{ 2 } = 60.297 ; ;
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