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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 90**2 = 125**2 + c**2 -2 * 90 * c * cos (45° ) ; ; ; ; c**2 -176.777c +7525 =0 ; ; p=1; q=-176.776695297; 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} = 88.3883476483 ± 16.9558249578 ; ; c_{1} = 105.344172606 ; ;
c_{2} = 71.4325226905 ; ; ; ; (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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 125**2-90**2-71.43**2 }{ 2 * 90 * 71.43 } ) = 100° 51'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-125**2-71.43**2 }{ 2 * 125 * 71.43 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 71.43**2-125**2-90**2 }{ 2 * 90 * 125 } ) = 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 ; ;




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