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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 45**2 = 72**2 + c**2 -2 * 45 * c * cos (28° ) ; ; ; ; c**2 -127.144c +3159 =0 ; ; p=1; q=-127.144453372; 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} = 63.5722266858 ± 29.7056897883 ; ; c_{1} = 93.2779164741 ; ;
c_{2} = 33.8665368976 ; ; ; ; (c -93.2779164741) (c -33.8665368976) = 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 = 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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