Triangle calculator SSA

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Triangle has two solutions with side c=93.27879164741 and with side c=33.86765368976

#1 Obtuse scalene triangle.

Sides: a = 72   b = 45   c = 93.27879164741

Area: T = 1576.488785194
Perimeter: p = 210.2787916474
Semiperimeter: s = 105.1398958237

Angle ∠ A = α = 48.6990483483° = 48°41'26″ = 0.85498092512 rad
Angle ∠ B = β = 28° = 0.48986921906 rad
Angle ∠ C = γ = 103.3109516517° = 103°18'34″ = 1.80330912119 rad

Height: ha = 43.79113292205
Height: hb = 70.06661267528
Height: hc = 33.80219525206

Median: ma = 63.77221322434
Median: mb = 80.22655249336
Median: mc = 37.80661843428

Inradius: r = 14.99443263503
Circumradius: R = 47.92662255343

Vertex coordinates: A[93.27879164741; 0] B[0; 0] C[63.57222266858; 33.80219525206]
Centroid: CG[52.28333810533; 11.26773175069]
Coordinates of the circumscribed circle: U[46.63989582371; -11.03331622178]
Coordinates of the inscribed circle: I[60.13989582371; 14.99443263503]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.3109516517° = 131°18'34″ = 0.85498092512 rad
∠ B' = β' = 152° = 0.48986921906 rad
∠ C' = γ' = 76.6990483483° = 76°41'26″ = 1.80330912119 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 72 ; ; b = 45 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 45**2 = 72**2 + c**2 -2 * 72 * c * cos (28° ) ; ; ; ; c**2 -127.144c +3159 =0 ; ; p=1; q=-127.144; r=3159 ; ; D = q**2 - 4pr = 127.144**2 - 4 * 1 * 3159 = 3529.71202318 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 127.14 ± sqrt{ 3529.71 } }{ 2 } ; ; c_{1,2} = 63.57222669 ± 29.7056897883 ; ; c_{1} = 93.2779164783 ; ;
c_{2} = 33.8665369017 ; ; ; ; (c -93.2779164783) (c -33.8665369017) = 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 = 72 ; ; b = 45 ; ; c = 93.28 ; ;

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

p = a+b+c = 72+45+93.28 = 210.28 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 210.28 }{ 2 } = 105.14 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 105.14 * (105.14-72)(105.14-45)(105.14-93.28) } ; ; T = sqrt{ 2485313.95 } = 1576.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1576.49 }{ 72 } = 43.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1576.49 }{ 45 } = 70.07 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1576.49 }{ 93.28 } = 33.8 ; ;

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{ 72**2-45**2-93.28**2 }{ 2 * 45 * 93.28 } ) = 48° 41'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 45**2-72**2-93.28**2 }{ 2 * 72 * 93.28 } ) = 28° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 93.28**2-72**2-45**2 }{ 2 * 45 * 72 } ) = 103° 18'34" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1576.49 }{ 105.14 } = 14.99 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 72 }{ 2 * sin 48° 41'26" } = 47.93 ; ;





#2 Obtuse scalene triangle.

Sides: a = 72   b = 45   c = 33.86765368976

Area: T = 572.3787536124
Perimeter: p = 150.8676536898
Semiperimeter: s = 75.43332684488

Angle ∠ A = α = 131.3109516517° = 131°18'34″ = 2.29217834024 rad
Angle ∠ B = β = 28° = 0.48986921906 rad
Angle ∠ C = γ = 20.6990483483° = 20°41'26″ = 0.36111170606 rad

Height: ha = 15.89993760035
Height: hb = 25.43990016055
Height: hc = 33.80219525206

Median: ma = 17.02985395944
Median: mb = 51.56876367572
Median: mc = 57.66000383649

Inradius: r = 7.58878660423
Circumradius: R = 47.92662255343

Vertex coordinates: A[33.86765368976; 0] B[0; 0] C[63.57222266858; 33.80219525206]
Centroid: CG[32.48795878611; 11.26773175069]
Coordinates of the circumscribed circle: U[16.93332684488; 44.83551147384]
Coordinates of the inscribed circle: I[30.43332684488; 7.58878660423]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 48.6990483483° = 48°41'26″ = 2.29217834024 rad
∠ B' = β' = 152° = 0.48986921906 rad
∠ C' = γ' = 159.3109516517° = 159°18'34″ = 0.36111170606 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 72 ; ; b = 45 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 45**2 = 72**2 + c**2 -2 * 72 * c * cos (28° ) ; ; ; ; c**2 -127.144c +3159 =0 ; ; p=1; q=-127.144; r=3159 ; ; D = q**2 - 4pr = 127.144**2 - 4 * 1 * 3159 = 3529.71202318 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 127.14 ± sqrt{ 3529.71 } }{ 2 } ; ; c_{1,2} = 63.57222669 ± 29.7056897883 ; ; c_{1} = 93.2779164783 ; ; : Nr. 1
c_{2} = 33.8665369017 ; ; ; ; (c -93.2779164783) (c -33.8665369017) = 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 = 72 ; ; b = 45 ; ; c = 33.87 ; ;

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

p = a+b+c = 72+45+33.87 = 150.87 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 150.87 }{ 2 } = 75.43 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 75.43 * (75.43-72)(75.43-45)(75.43-33.87) } ; ; T = sqrt{ 327616.04 } = 572.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 572.38 }{ 72 } = 15.9 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 572.38 }{ 45 } = 25.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 572.38 }{ 33.87 } = 33.8 ; ;

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{ 72**2-45**2-33.87**2 }{ 2 * 45 * 33.87 } ) = 131° 18'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 45**2-72**2-33.87**2 }{ 2 * 72 * 33.87 } ) = 28° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 33.87**2-72**2-45**2 }{ 2 * 45 * 72 } ) = 20° 41'26" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 572.38 }{ 75.43 } = 7.59 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 72 }{ 2 * sin 131° 18'34" } = 47.93 ; ;




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