Triangle calculator SSA

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Triangle has two solutions with side c=25.4111158487 and with side c=9.30442157098

#1 Obtuse scalene triangle.

Sides: a = 17.97   b = 9.3   c = 25.4111158487

Area: T = 59.09333725943
Perimeter: p = 52.6811158487
Semiperimeter: s = 26.34105792435

Angle ∠ A = α = 30.00769593698° = 30°25″ = 0.52437202395 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 134.993304063° = 134°59'35″ = 2.35660730263 rad

Height: ha = 6.57768917745
Height: hb = 12.70882521708
Height: hc = 4.65109782405

Median: ma = 16.89331424793
Median: mb = 21.51104959921
Median: mc = 6.57882753125

Inradius: r = 2.24334348177
Circumradius: R = 17.9666220369

Vertex coordinates: A[25.4111158487; 0] B[0; 0] C[17.35876870984; 4.65109782405]
Centroid: CG[14.25662818618; 1.55503260802]
Coordinates of the circumscribed circle: U[12.70655792435; -12.70224930795]
Coordinates of the inscribed circle: I[17.04105792435; 2.24334348177]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.993304063° = 149°59'35″ = 0.52437202395 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 45.00769593698° = 45°25″ = 2.35660730263 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 17.97 ; ; b = 9.3 ; ; beta = 15° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 9.3**2 = 17.97**2 + c**2 -2 * 17.97 * c * cos (15° ) ; ; ; ; c**2 -34.715c +236.431 =0 ; ; p=1; q=-34.715; r=236.431 ; ; D = q**2 - 4pr = 34.715**2 - 4 * 1 * 236.431 = 259.433605626 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 34.72 ± sqrt{ 259.43 } }{ 2 } ; ; c_{1,2} = 17.3576871 ± 8.05347138857 ; ; c_{1} = 25.4111584886 ; ;
c_{2} = 9.30421571143 ; ; ; ; text{ Factored form: } ; ; (c -25.4111584886) (c -9.30421571143) = 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 = 17.97 ; ; b = 9.3 ; ; c = 25.41 ; ;

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

p = a+b+c = 17.97+9.3+25.41 = 52.68 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 52.68 }{ 2 } = 26.34 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.34 * (26.34-17.97)(26.34-9.3)(26.34-25.41) } ; ; T = sqrt{ 3492.03 } = 59.09 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 59.09 }{ 17.97 } = 6.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 59.09 }{ 9.3 } = 12.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 59.09 }{ 25.41 } = 4.65 ; ;

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{ 9.3**2+25.41**2-17.97**2 }{ 2 * 9.3 * 25.41 } ) = 30° 25" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 17.97**2+25.41**2-9.3**2 }{ 2 * 17.97 * 25.41 } ) = 15° ; ; gamma = 180° - alpha - beta = 180° - 30° 25" - 15° = 134° 59'35" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 59.09 }{ 26.34 } = 2.24 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 17.97 }{ 2 * sin 30° 25" } = 17.97 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.3**2+2 * 25.41**2 - 17.97**2 } }{ 2 } = 16.893 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 25.41**2+2 * 17.97**2 - 9.3**2 } }{ 2 } = 21.51 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.3**2+2 * 17.97**2 - 25.41**2 } }{ 2 } = 6.578 ; ;







#2 Obtuse scalene triangle.

Sides: a = 17.97   b = 9.3   c = 9.30442157098

Area: T = 21.63768524057
Perimeter: p = 36.57442157098
Semiperimeter: s = 18.28771078549

Angle ∠ A = α = 149.993304063° = 149°59'35″ = 2.61878724141 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 15.00769593698° = 15°25″ = 0.26219208517 rad

Height: ha = 2.40881082255
Height: hb = 4.65330865389
Height: hc = 4.65109782405

Median: ma = 2.40881092142
Median: mb = 13.5322263853
Median: mc = 13.53300902623

Inradius: r = 1.18331751952
Circumradius: R = 17.9666220369

Vertex coordinates: A[9.30442157098; 0] B[0; 0] C[17.35876870984; 4.65109782405]
Centroid: CG[8.88773009361; 1.55503260802]
Coordinates of the circumscribed circle: U[4.65221078549; 17.353347132]
Coordinates of the inscribed circle: I[8.98771078549; 1.18331751952]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 30.00769593698° = 30°25″ = 2.61878724141 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 164.993304063° = 164°59'35″ = 0.26219208517 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 17.97 ; ; b = 9.3 ; ; beta = 15° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 9.3**2 = 17.97**2 + c**2 -2 * 17.97 * c * cos (15° ) ; ; ; ; c**2 -34.715c +236.431 =0 ; ; p=1; q=-34.715; r=236.431 ; ; D = q**2 - 4pr = 34.715**2 - 4 * 1 * 236.431 = 259.433605626 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 34.72 ± sqrt{ 259.43 } }{ 2 } ; ; c_{1,2} = 17.3576871 ± 8.05347138857 ; ; c_{1} = 25.4111584886 ; ; : Nr. 1
c_{2} = 9.30421571143 ; ; ; ; text{ Factored form: } ; ; (c -25.4111584886) (c -9.30421571143) = 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 = 17.97 ; ; b = 9.3 ; ; c = 9.3 ; ;

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

p = a+b+c = 17.97+9.3+9.3 = 36.57 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 36.57 }{ 2 } = 18.29 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.29 * (18.29-17.97)(18.29-9.3)(18.29-9.3) } ; ; T = sqrt{ 468.15 } = 21.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21.64 }{ 17.97 } = 2.41 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21.64 }{ 9.3 } = 4.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21.64 }{ 9.3 } = 4.65 ; ;

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{ 9.3**2+9.3**2-17.97**2 }{ 2 * 9.3 * 9.3 } ) = 149° 59'35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 17.97**2+9.3**2-9.3**2 }{ 2 * 17.97 * 9.3 } ) = 15° ; ; gamma = 180° - alpha - beta = 180° - 149° 59'35" - 15° = 15° 25" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21.64 }{ 18.29 } = 1.18 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 17.97 }{ 2 * sin 149° 59'35" } = 17.97 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.3**2+2 * 9.3**2 - 17.97**2 } }{ 2 } = 2.408 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.3**2+2 * 17.97**2 - 9.3**2 } }{ 2 } = 13.532 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.3**2+2 * 17.97**2 - 9.3**2 } }{ 2 } = 13.53 ; ;
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