Triangle calculator SSA

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Triangle has two solutions with side c=70.40987357372 and with side c=8.45106559274

#1 Obtuse scalene triangle.

Sides: a = 46   b = 39   c = 70.40987357372

Area: T = 834.0533133351
Perimeter: p = 155.4098735737
Semiperimeter: s = 77.70443678686

Angle ∠ A = α = 37.40875709154° = 37°24'27″ = 0.65328852776 rad
Angle ∠ B = β = 31° = 0.54110520681 rad
Angle ∠ C = γ = 111.5922429085° = 111°35'33″ = 1.94876553078 rad

Height: ha = 36.26331797109
Height: hb = 42.77219555565
Height: hc = 23.69217514459

Median: ma = 52.06595335559
Median: mb = 56.18222483891
Median: mc = 24.06655871105

Inradius: r = 10.73436711723
Circumradius: R = 37.8611278515

Vertex coordinates: A[70.40987357372; 0] B[0; 0] C[39.43296958323; 23.69217514459]
Centroid: CG[36.61328105232; 7.8977250482]
Coordinates of the circumscribed circle: U[35.20443678686; -13.93330145254]
Coordinates of the inscribed circle: I[38.70443678686; 10.73436711723]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.5922429085° = 142°35'33″ = 0.65328852776 rad
∠ B' = β' = 149° = 0.54110520681 rad
∠ C' = γ' = 68.40875709154° = 68°24'27″ = 1.94876553078 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 = 46 ; ; b = 39 ; ; c = 70.41 ; ;

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

p = a+b+c = 46+39+70.41 = 155.41 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 155.41 }{ 2 } = 77.7 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 77.7 * (77.7-46)(77.7-39)(77.7-70.41) } ; ; T = sqrt{ 695644.63 } = 834.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 834.05 }{ 46 } = 36.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 834.05 }{ 39 } = 42.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 834.05 }{ 70.41 } = 23.69 ; ;

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{ 46**2-39**2-70.41**2 }{ 2 * 39 * 70.41 } ) = 37° 24'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 39**2-46**2-70.41**2 }{ 2 * 46 * 70.41 } ) = 31° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 70.41**2-46**2-39**2 }{ 2 * 39 * 46 } ) = 111° 35'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 834.05 }{ 77.7 } = 10.73 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 46 }{ 2 * sin 37° 24'27" } = 37.86 ; ;





#2 Obtuse scalene triangle.

Sides: a = 46   b = 39   c = 8.45106559274

Area: T = 100.1055419894
Perimeter: p = 93.45106559274
Semiperimeter: s = 46.72553279637

Angle ∠ A = α = 142.5922429085° = 142°35'33″ = 2.48987073759 rad
Angle ∠ B = β = 31° = 0.54110520681 rad
Angle ∠ C = γ = 6.40875709154° = 6°24'27″ = 0.11218332095 rad

Height: ha = 4.35224095606
Height: hb = 5.13436112766
Height: hc = 23.69217514459

Median: ma = 16.34664611706
Median: mb = 26.71106119885
Median: mc = 42.43440264835

Inradius: r = 2.14224230552
Circumradius: R = 37.8611278515

Vertex coordinates: A[8.45106559274; 0] B[0; 0] C[39.43296958323; 23.69217514459]
Centroid: CG[15.96601172532; 7.8977250482]
Coordinates of the circumscribed circle: U[4.22553279637; 37.62547659712]
Coordinates of the inscribed circle: I[7.72553279637; 2.14224230552]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 37.40875709154° = 37°24'27″ = 2.48987073759 rad
∠ B' = β' = 149° = 0.54110520681 rad
∠ C' = γ' = 173.5922429085° = 173°35'33″ = 0.11218332095 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 46 ; ; b = 39 ; ; beta = 31° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 39**2 = 46**2 + c**2 -2 * 39 * c * cos (31° ) ; ; ; ; c**2 -78.859c +595 =0 ; ; p=1; q=-78.8593916646; r=595 ; ; D = q**2 - 4pr = 78.859**2 - 4 * 1 * 595 = 3838.80365371 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 78.86 ± sqrt{ 3838.8 } }{ 2 } ; ; c_{1,2} = 39.4296958323 ± 30.9790399049 ; ; c_{1} = 70.4087357372 ; ;
c_{2} = 8.45065592743 ; ; ; ; (c -70.4087357372) (c -8.45065592743) = 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 = 46 ; ; b = 39 ; ; c = 8.45 ; ;

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

p = a+b+c = 46+39+8.45 = 93.45 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 93.45 }{ 2 } = 46.73 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 46.73 * (46.73-46)(46.73-39)(46.73-8.45) } ; ; T = sqrt{ 10021.1 } = 100.11 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 100.11 }{ 46 } = 4.35 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 100.11 }{ 39 } = 5.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 100.11 }{ 8.45 } = 23.69 ; ;

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{ 46**2-39**2-8.45**2 }{ 2 * 39 * 8.45 } ) = 142° 35'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 39**2-46**2-8.45**2 }{ 2 * 46 * 8.45 } ) = 31° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.45**2-46**2-39**2 }{ 2 * 39 * 46 } ) = 6° 24'27" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 100.11 }{ 46.73 } = 2.14 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 46 }{ 2 * sin 142° 35'33" } = 37.86 ; ;




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