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?

1. Use Law of Cosines

a = 46 ; ; b = 39 ; ; beta = 31° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 39**2 = 46**2 + c**2 -2 * 46 * c * cos (31° ) ; ; ; ; c**2 -78.859c +595 =0 ; ; p=1; q=-78.859; 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} = 39.42969583 ± 30.9790399049 ; ; c_{1} = 70.4087357372 ; ; c_{2} = 8.45065592743 ; ; ; ; text{ Factored form: } ; ; (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 = 70.41 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 39**2+70.41**2-46**2 }{ 2 * 39 * 70.41 } ) = 37° 24'27" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 46**2+70.41**2-39**2 }{ 2 * 46 * 70.41 } ) = 31° ; ;
 gamma = 180° - alpha - beta = 180° - 37° 24'27" - 31° = 111° 35'33" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 39**2+2 * 70.41**2 - 46**2 } }{ 2 } = 52.06 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 70.41**2+2 * 46**2 - 39**2 } }{ 2 } = 56.182 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 39**2+2 * 46**2 - 70.41**2 } }{ 2 } = 24.066 ; ;



#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 - 2ac cos beta ; ; 39**2 = 46**2 + c**2 -2 * 46 * c * cos (31° ) ; ; ; ; c**2 -78.859c +595 =0 ; ; p=1; q=-78.859; 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 } ; ; : Nr. 1
c_{1,2} = 39.42969583 ± 30.9790399049 ; ; c_{1} = 70.4087357372 ; ; c_{2} = 8.45065592743 ; ; ; ; text{ Factored form: } ; ; (c -70.4087357372) (c -8.45065592743) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 39**2+8.45**2-46**2 }{ 2 * 39 * 8.45 } ) = 142° 35'33" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 46**2+8.45**2-39**2 }{ 2 * 46 * 8.45 } ) = 31° ; ;
 gamma = 180° - alpha - beta = 180° - 142° 35'33" - 31° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 39**2+2 * 8.45**2 - 46**2 } }{ 2 } = 16.346 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.45**2+2 * 46**2 - 39**2 } }{ 2 } = 26.711 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 39**2+2 * 46**2 - 8.45**2 } }{ 2 } = 42.434 ; ;
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