Triangle calculator SSA

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Triangle has two solutions with side c=321.4222498361 and with side c=7.47699189143

#1 Obtuse scalene triangle.

Sides: a = 175   b = 168   c = 321.4222498361

Area: T = 9619.135478378
Perimeter: p = 664.4222498361
Semiperimeter: s = 332.211124918

Angle ∠ A = α = 20.87113600892° = 20°52'17″ = 0.36442739529 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 139.1298639911° = 139°7'43″ = 2.42882528503 rad

Height: ha = 109.9332968957
Height: hb = 114.5143509331
Height: hc = 59.8543525082

Median: ma = 241.0644226351
Median: mb = 244.7710731964
Median: mc = 59.97699456969

Inradius: r = 28.95548737663
Circumradius: R = 245.6599569614

Vertex coordinates: A[321.4222498361; 0] B[0; 0] C[164.4466208638; 59.8543525082]
Centroid: CG[161.9566235666; 19.95111750273]
Coordinates of the circumscribed circle: U[160.711124918; -185.7187643161]
Coordinates of the inscribed circle: I[164.211124918; 28.95548737663]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.1298639911° = 159°7'43″ = 0.36442739529 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 40.87113600892° = 40°52'17″ = 2.42882528503 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 175 ; ; b = 168 ; ; beta = 20° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 168**2 = 175**2 + c**2 -2 * 175 * c * cos (20° ) ; ; ; ; c**2 -328.892c +2401 =0 ; ; p=1; q=-328.892; r=2401 ; ; D = q**2 - 4pr = 328.892**2 - 4 * 1 * 2401 = 98566.222141 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 328.89 ± sqrt{ 98566.22 } }{ 2 } ; ;
c_{1,2} = 164.44620864 ± 156.976289723 ; ; c_{1} = 321.422498361 ; ; c_{2} = 7.46991891434 ; ; ; ; text{ Factored form: } ; ; (c -321.422498361) (c -7.46991891434) = 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 = 175 ; ; b = 168 ; ; c = 321.42 ; ;

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

p = a+b+c = 175+168+321.42 = 664.42 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 664.42 }{ 2 } = 332.21 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 332.21 * (332.21-175)(332.21-168)(332.21-321.42) } ; ; T = sqrt{ 92527753.99 } = 9619.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9619.13 }{ 175 } = 109.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9619.13 }{ 168 } = 114.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9619.13 }{ 321.42 } = 59.85 ; ;

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{ 168**2+321.42**2-175**2 }{ 2 * 168 * 321.42 } ) = 20° 52'17" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 175**2+321.42**2-168**2 }{ 2 * 175 * 321.42 } ) = 20° ; ;
 gamma = 180° - alpha - beta = 180° - 20° 52'17" - 20° = 139° 7'43" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9619.13 }{ 332.21 } = 28.95 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 175 }{ 2 * sin 20° 52'17" } = 245.6 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 168**2+2 * 321.42**2 - 175**2 } }{ 2 } = 241.064 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 321.42**2+2 * 175**2 - 168**2 } }{ 2 } = 244.771 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 168**2+2 * 175**2 - 321.42**2 } }{ 2 } = 59.97 ; ;



#2 Obtuse scalene triangle.

Sides: a = 175   b = 168   c = 7.47699189143

Area: T = 223.555048955
Perimeter: p = 350.4769918914
Semiperimeter: s = 175.2354959457

Angle ∠ A = α = 159.1298639911° = 159°7'43″ = 2.77773187007 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 0.87113600892° = 0°52'17″ = 0.01552081025 rad

Height: ha = 2.55548627377
Height: hb = 2.66113153518
Height: hc = 59.8543525082

Median: ma = 80.52111142763
Median: mb = 91.01986785462
Median: mc = 171.4955043887

Inradius: r = 1.27657185566
Circumradius: R = 245.6599569614

Vertex coordinates: A[7.47699189143; 0] B[0; 0] C[164.4466208638; 59.8543525082]
Centroid: CG[57.30553758506; 19.95111750273]
Coordinates of the circumscribed circle: U[3.73549594572; 245.5711168243]
Coordinates of the inscribed circle: I[7.23549594572; 1.27657185566]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 20.87113600892° = 20°52'17″ = 2.77773187007 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 179.1298639911° = 179°7'43″ = 0.01552081025 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 175 ; ; b = 168 ; ; beta = 20° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 168**2 = 175**2 + c**2 -2 * 175 * c * cos (20° ) ; ; ; ; c**2 -328.892c +2401 =0 ; ; p=1; q=-328.892; r=2401 ; ; D = q**2 - 4pr = 328.892**2 - 4 * 1 * 2401 = 98566.222141 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 328.89 ± sqrt{ 98566.22 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 164.44620864 ± 156.976289723 ; ; c_{1} = 321.422498361 ; ; c_{2} = 7.46991891434 ; ; ; ; text{ Factored form: } ; ; (c -321.422498361) (c -7.46991891434) = 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 = 175 ; ; b = 168 ; ; c = 7.47 ; ;

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

p = a+b+c = 175+168+7.47 = 350.47 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 350.47 }{ 2 } = 175.23 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 175.23 * (175.23-175)(175.23-168)(175.23-7.47) } ; ; T = sqrt{ 49974.82 } = 223.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 223.55 }{ 175 } = 2.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 223.55 }{ 168 } = 2.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 223.55 }{ 7.47 } = 59.85 ; ;

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{ 168**2+7.47**2-175**2 }{ 2 * 168 * 7.47 } ) = 159° 7'43" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 175**2+7.47**2-168**2 }{ 2 * 175 * 7.47 } ) = 20° ; ;
 gamma = 180° - alpha - beta = 180° - 159° 7'43" - 20° = 0° 52'17" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 223.55 }{ 175.23 } = 1.28 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 175 }{ 2 * sin 159° 7'43" } = 245.6 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 168**2+2 * 7.47**2 - 175**2 } }{ 2 } = 80.521 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.47**2+2 * 175**2 - 168**2 } }{ 2 } = 91.019 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 168**2+2 * 175**2 - 7.47**2 } }{ 2 } = 171.495 ; ;
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