Triangle calculator SSA

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Triangle has two solutions with side c=28.55661452939 and with side c=17.56604058195

#1 Obtuse scalene triangle.

Sides: a = 23.6   b = 7.45   c = 28.55661452939

Area: T = 71.78332546173
Perimeter: p = 59.60661452939
Semiperimeter: s = 29.80330726469

Angle ∠ A = α = 42.44113004475° = 42°26'29″ = 0.74107404316 rad
Angle ∠ B = β = 12.3° = 12°18' = 0.2154675498 rad
Angle ∠ C = γ = 125.2598699552° = 125°15'31″ = 2.1866176724 rad

Height: ha = 6.08333266625
Height: hb = 19.27106723805
Height: hc = 5.02875171161

Median: ma = 17.21215649208
Median: mb = 25.92993480833
Median: mc = 10.11877018877

Inradius: r = 2.40985857008
Circumradius: R = 17.48657684161

Vertex coordinates: A[28.55661452939; 0] B[0; 0] C[23.05882755567; 5.02875171161]
Centroid: CG[17.20548069502; 1.67658390387]
Coordinates of the circumscribed circle: U[14.27880726469; -10.09439951749]
Coordinates of the inscribed circle: I[22.35330726469; 2.40985857008]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.5598699552° = 137°33'31″ = 0.74107404316 rad
∠ B' = β' = 167.7° = 167°42' = 0.2154675498 rad
∠ C' = γ' = 54.74113004475° = 54°44'29″ = 2.1866176724 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 23.6 ; ; b = 7.45 ; ; beta = 12° 18' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 7.45**2 = 23.6**2 + c**2 -2 * 23.6 * c * cos (12° 18') ; ; ; ; c**2 -46.117c +501.458 =0 ; ; p=1; q=-46.117; r=501.458 ; ; D = q**2 - 4pr = 46.117**2 - 4 * 1 * 501.458 = 120.90628659 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 46.12 ± sqrt{ 120.91 } }{ 2 } ; ; c_{1,2} = 23.05827556 ± 5.49786973722 ; ;
c_{1} = 28.5561452972 ; ; c_{2} = 17.5604058228 ; ; ; ; text{ Factored form: } ; ; (c -28.5561452972) (c -17.5604058228) = 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 = 23.6 ; ; b = 7.45 ; ; c = 28.56 ; ;

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

p = a+b+c = 23.6+7.45+28.56 = 59.61 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59.61 }{ 2 } = 29.8 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.8 * (29.8-23.6)(29.8-7.45)(29.8-28.56) } ; ; T = sqrt{ 5152.84 } = 71.78 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 71.78 }{ 23.6 } = 6.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 71.78 }{ 7.45 } = 19.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 71.78 }{ 28.56 } = 5.03 ; ;

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{ 7.45**2+28.56**2-23.6**2 }{ 2 * 7.45 * 28.56 } ) = 42° 26'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 23.6**2+28.56**2-7.45**2 }{ 2 * 23.6 * 28.56 } ) = 12° 18' ; ; gamma = 180° - alpha - beta = 180° - 42° 26'29" - 12° 18' = 125° 15'31" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 71.78 }{ 29.8 } = 2.41 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 23.6 }{ 2 * sin 42° 26'29" } = 17.49 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.45**2+2 * 28.56**2 - 23.6**2 } }{ 2 } = 17.212 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 28.56**2+2 * 23.6**2 - 7.45**2 } }{ 2 } = 25.929 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.45**2+2 * 23.6**2 - 28.56**2 } }{ 2 } = 10.118 ; ;







#2 Obtuse scalene triangle.

Sides: a = 23.6   b = 7.45   c = 17.56604058195

Area: T = 44.14326204114
Perimeter: p = 48.61104058195
Semiperimeter: s = 24.30552029097

Angle ∠ A = α = 137.5598699552° = 137°33'31″ = 2.4010852222 rad
Angle ∠ B = β = 12.3° = 12°18' = 0.2154675498 rad
Angle ∠ C = γ = 30.14113004475° = 30°8'29″ = 0.52660649336 rad

Height: ha = 3.74109000349
Height: hb = 11.85503678957
Height: hc = 5.02875171161

Median: ma = 6.5344154595
Median: mb = 20.46443177573
Median: mc = 15.13773474184

Inradius: r = 1.81661798762
Circumradius: R = 17.48657684161

Vertex coordinates: A[17.56604058195; 0] B[0; 0] C[23.05882755567; 5.02875171161]
Centroid: CG[13.54395604587; 1.67658390387]
Coordinates of the circumscribed circle: U[8.78802029097; 15.1221512291]
Coordinates of the inscribed circle: I[16.85552029097; 1.81661798762]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 42.44113004475° = 42°26'29″ = 2.4010852222 rad
∠ B' = β' = 167.7° = 167°42' = 0.2154675498 rad
∠ C' = γ' = 149.8598699552° = 149°51'31″ = 0.52660649336 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 23.6 ; ; b = 7.45 ; ; beta = 12° 18' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 7.45**2 = 23.6**2 + c**2 -2 * 23.6 * c * cos (12° 18') ; ; ; ; c**2 -46.117c +501.458 =0 ; ; p=1; q=-46.117; r=501.458 ; ; D = q**2 - 4pr = 46.117**2 - 4 * 1 * 501.458 = 120.90628659 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 46.12 ± sqrt{ 120.91 } }{ 2 } ; ; c_{1,2} = 23.05827556 ± 5.49786973722 ; ; : Nr. 1
c_{1} = 28.5561452972 ; ; c_{2} = 17.5604058228 ; ; ; ; text{ Factored form: } ; ; (c -28.5561452972) (c -17.5604058228) = 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 = 23.6 ; ; b = 7.45 ; ; c = 17.56 ; ;

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

p = a+b+c = 23.6+7.45+17.56 = 48.61 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 48.61 }{ 2 } = 24.31 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.31 * (24.31-23.6)(24.31-7.45)(24.31-17.56) } ; ; T = sqrt{ 1948.57 } = 44.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 44.14 }{ 23.6 } = 3.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 44.14 }{ 7.45 } = 11.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 44.14 }{ 17.56 } = 5.03 ; ;

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{ 7.45**2+17.56**2-23.6**2 }{ 2 * 7.45 * 17.56 } ) = 137° 33'31" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 23.6**2+17.56**2-7.45**2 }{ 2 * 23.6 * 17.56 } ) = 12° 18' ; ; gamma = 180° - alpha - beta = 180° - 137° 33'31" - 12° 18' = 30° 8'29" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 44.14 }{ 24.31 } = 1.82 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 23.6 }{ 2 * sin 137° 33'31" } = 17.49 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.45**2+2 * 17.56**2 - 23.6**2 } }{ 2 } = 6.534 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.56**2+2 * 23.6**2 - 7.45**2 } }{ 2 } = 20.464 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.45**2+2 * 23.6**2 - 17.56**2 } }{ 2 } = 15.137 ; ;
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