Triangle calculator SSA

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Triangle has two solutions with side c=105.3444172606 and with side c=71.43325226905

#1 Acute scalene triangle.

Sides: a = 125   b = 90   c = 105.3444172606

Area: T = 4655.599867552
Perimeter: p = 320.3444172606
Semiperimeter: s = 160.1722086303

Angle ∠ A = α = 79.1410688732° = 79°8'26″ = 1.38112655907 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 55.8599311268° = 55°51'34″ = 0.97549288995 rad

Height: ha = 74.49895788083
Height: hb = 103.4587748345
Height: hc = 88.38883476483

Median: ma = 75.4488309133
Median: mb = 106.4721580016
Median: mc = 95.33217959785

Inradius: r = 29.06662298467
Circumradius: R = 63.64396103068

Vertex coordinates: A[105.3444172606; 0] B[0; 0] C[88.38883476483; 88.38883476483]
Centroid: CG[64.57875067515; 29.46327825494]
Coordinates of the circumscribed circle: U[52.67220863031; 35.71662613453]
Coordinates of the inscribed circle: I[70.17220863031; 29.06662298467]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 100.8599311268° = 100°51'34″ = 1.38112655907 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 124.1410688732° = 124°8'26″ = 0.97549288995 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 125 ; ; b = 90 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 125**2 + c**2 -2 * 125 * c * cos (45° ) ; ; ; ; c**2 -176.777c +7525 =0 ; ; p=1; q=-176.777; r=7525 ; ; D = q**2 - 4pr = 176.777**2 - 4 * 1 * 7525 = 1150 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 176.78 ± sqrt{ 1150 } }{ 2 } ; ;
c_{1,2} = 88.38834765 ± 16.9558249578 ; ; c_{1} = 105.344172606 ; ; c_{2} = 71.4325226905 ; ; ; ; text{ Factored form: } ; ; (c -105.344172606) (c -71.4325226905) = 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 = 125 ; ; b = 90 ; ; c = 105.34 ; ;

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

p = a+b+c = 125+90+105.34 = 320.34 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 320.34 }{ 2 } = 160.17 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 160.17 * (160.17-125)(160.17-90)(160.17-105.34) } ; ; T = sqrt{ 21674599.03 } = 4655.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4655.6 }{ 125 } = 74.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4655.6 }{ 90 } = 103.46 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4655.6 }{ 105.34 } = 88.39 ; ;

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{ 90**2+105.34**2-125**2 }{ 2 * 90 * 105.34 } ) = 79° 8'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 125**2+105.34**2-90**2 }{ 2 * 125 * 105.34 } ) = 45° ; ;
 gamma = 180° - alpha - beta = 180° - 79° 8'26" - 45° = 55° 51'34" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4655.6 }{ 160.17 } = 29.07 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 125 }{ 2 * sin 79° 8'26" } = 63.64 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 105.34**2 - 125**2 } }{ 2 } = 75.448 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 105.34**2+2 * 125**2 - 90**2 } }{ 2 } = 106.472 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 125**2 - 105.34**2 } }{ 2 } = 95.332 ; ;



#2 Obtuse scalene triangle.

Sides: a = 125   b = 90   c = 71.43325226905

Area: T = 3156.901132448
Perimeter: p = 286.433252269
Semiperimeter: s = 143.2166261345

Angle ∠ A = α = 100.8599311268° = 100°51'34″ = 1.76603270629 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 34.1410688732° = 34°8'26″ = 0.59658674273 rad

Height: ha = 50.51104211917
Height: hb = 70.15333627663
Height: hc = 88.38883476483

Median: ma = 51.91438964918
Median: mb = 91.31770446793
Median: mc = 102.892241311

Inradius: r = 22.0432897188
Circumradius: R = 63.64396103068

Vertex coordinates: A[71.43325226905; 0] B[0; 0] C[88.38883476483; 88.38883476483]
Centroid: CG[53.27436234463; 29.46327825494]
Coordinates of the circumscribed circle: U[35.71662613453; 52.67220863031]
Coordinates of the inscribed circle: I[53.21662613453; 22.0432897188]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 79.1410688732° = 79°8'26″ = 1.76603270629 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 145.8599311268° = 145°51'34″ = 0.59658674273 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 125 ; ; b = 90 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 125**2 + c**2 -2 * 125 * c * cos (45° ) ; ; ; ; c**2 -176.777c +7525 =0 ; ; p=1; q=-176.777; r=7525 ; ; D = q**2 - 4pr = 176.777**2 - 4 * 1 * 7525 = 1150 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 176.78 ± sqrt{ 1150 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 88.38834765 ± 16.9558249578 ; ; c_{1} = 105.344172606 ; ; c_{2} = 71.4325226905 ; ; ; ; text{ Factored form: } ; ; (c -105.344172606) (c -71.4325226905) = 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 = 125 ; ; b = 90 ; ; c = 71.43 ; ;

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

p = a+b+c = 125+90+71.43 = 286.43 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 286.43 }{ 2 } = 143.22 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 143.22 * (143.22-125)(143.22-90)(143.22-71.43) } ; ; T = sqrt{ 9966025.97 } = 3156.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3156.9 }{ 125 } = 50.51 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3156.9 }{ 90 } = 70.15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3156.9 }{ 71.43 } = 88.39 ; ;

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{ 90**2+71.43**2-125**2 }{ 2 * 90 * 71.43 } ) = 100° 51'34" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 125**2+71.43**2-90**2 }{ 2 * 125 * 71.43 } ) = 45° ; ;
 gamma = 180° - alpha - beta = 180° - 100° 51'34" - 45° = 34° 8'26" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3156.9 }{ 143.22 } = 22.04 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 125 }{ 2 * sin 100° 51'34" } = 63.64 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 71.43**2 - 125**2 } }{ 2 } = 51.914 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 71.43**2+2 * 125**2 - 90**2 } }{ 2 } = 91.317 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 125**2 - 71.43**2 } }{ 2 } = 102.892 ; ;
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