Triangle calculator SSA

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Triangle has two solutions with side c=21.43296434539 and with side c=2.95503244016

#1 Obtuse scalene triangle.

Sides: a = 12.62   b = 9.8   c = 21.43296434539

Area: T = 34.9987783089
Perimeter: p = 43.85496434539
Semiperimeter: s = 21.9254821727

Angle ∠ A = α = 19.46989242535° = 19°28'8″ = 0.343979683 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 145.5311075747° = 145°31'52″ = 2.54399964357 rad

Height: ha = 5.54663998556
Height: hb = 7.1422404712
Height: hc = 3.26662963492

Median: ma = 15.42113718353
Median: mb = 16.88989611664
Median: mc = 3.58439636381

Inradius: r = 1.59662630632
Circumradius: R = 18.93221461953

Vertex coordinates: A[21.43296434539; 0] B[0; 0] C[12.19899839278; 3.26662963492]
Centroid: CG[11.20765424606; 1.08987654497]
Coordinates of the circumscribed circle: U[10.7154821727; -15.60882912235]
Coordinates of the inscribed circle: I[12.1254821727; 1.59662630632]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.5311075747° = 160°31'52″ = 0.343979683 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 34.46989242535° = 34°28'8″ = 2.54399964357 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 12.62 ; ; b = 9.8 ; ; beta = 15° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 9.8**2 = 12.62**2 + c**2 -2 * 12.62 * c * cos (15° ) ; ; ; ; c**2 -24.38c +63.224 =0 ; ; p=1; q=-24.38; r=63.224 ; ; D = q**2 - 4pr = 24.38**2 - 4 * 1 * 63.224 = 341.485232637 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 24.38 ± sqrt{ 341.49 } }{ 2 } ; ; c_{1,2} = 12.18998393 ± 9.23965952615 ; ; c_{1} = 21.4296434562 ; ;
c_{2} = 2.95032440385 ; ; ; ; text{ Factored form: } ; ; (c -21.4296434562) (c -2.95032440385) = 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 = 12.62 ; ; b = 9.8 ; ; c = 21.43 ; ;

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

p = a+b+c = 12.62+9.8+21.43 = 43.85 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43.85 }{ 2 } = 21.92 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.92 * (21.92-12.62)(21.92-9.8)(21.92-21.43) } ; ; T = sqrt{ 1224.84 } = 35 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 35 }{ 12.62 } = 5.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 35 }{ 9.8 } = 7.14 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 35 }{ 21.43 } = 3.27 ; ;

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{ 9.8**2+21.43**2-12.62**2 }{ 2 * 9.8 * 21.43 } ) = 19° 28'8" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.62**2+21.43**2-9.8**2 }{ 2 * 12.62 * 21.43 } ) = 15° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 12.62**2+9.8**2-21.43**2 }{ 2 * 12.62 * 9.8 } ) = 145° 31'52" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 35 }{ 21.92 } = 1.6 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.62 }{ 2 * sin 19° 28'8" } = 18.93 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.8**2+2 * 21.43**2 - 12.62**2 } }{ 2 } = 15.421 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.43**2+2 * 12.62**2 - 9.8**2 } }{ 2 } = 16.889 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.8**2+2 * 12.62**2 - 21.43**2 } }{ 2 } = 3.584 ; ;







#2 Obtuse scalene triangle.

Sides: a = 12.62   b = 9.8   c = 2.95503244016

Area: T = 4.8188316911
Perimeter: p = 25.37703244016
Semiperimeter: s = 12.68551622008

Angle ∠ A = α = 160.5311075747° = 160°31'52″ = 2.80217958235 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 4.46989242535° = 4°28'8″ = 0.07879974422 rad

Height: ha = 0.76436001444
Height: hb = 0.98333299818
Height: hc = 3.26662963492

Median: ma = 3.54334597553
Median: mb = 7.74443144976
Median: mc = 11.20216113341

Inradius: r = 0.38798388097
Circumradius: R = 18.93221461953

Vertex coordinates: A[2.95503244016; 0] B[0; 0] C[12.19899839278; 3.26662963492]
Centroid: CG[5.04767694431; 1.08987654497]
Coordinates of the circumscribed circle: U[1.47551622008; 18.87545875727]
Coordinates of the inscribed circle: I[2.88551622008; 0.38798388097]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 19.46989242535° = 19°28'8″ = 2.80217958235 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 175.5311075747° = 175°31'52″ = 0.07879974422 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 12.62 ; ; b = 9.8 ; ; beta = 15° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 9.8**2 = 12.62**2 + c**2 -2 * 12.62 * c * cos (15° ) ; ; ; ; c**2 -24.38c +63.224 =0 ; ; p=1; q=-24.38; r=63.224 ; ; D = q**2 - 4pr = 24.38**2 - 4 * 1 * 63.224 = 341.485232637 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 24.38 ± sqrt{ 341.49 } }{ 2 } ; ; c_{1,2} = 12.18998393 ± 9.23965952615 ; ; c_{1} = 21.4296434562 ; ; : Nr. 1
c_{2} = 2.95032440385 ; ; ; ; text{ Factored form: } ; ; (c -21.4296434562) (c -2.95032440385) = 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 = 12.62 ; ; b = 9.8 ; ; c = 2.95 ; ;

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

p = a+b+c = 12.62+9.8+2.95 = 25.37 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25.37 }{ 2 } = 12.69 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.69 * (12.69-12.62)(12.69-9.8)(12.69-2.95) } ; ; T = sqrt{ 23.22 } = 4.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.82 }{ 12.62 } = 0.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.82 }{ 9.8 } = 0.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.82 }{ 2.95 } = 3.27 ; ;

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{ 9.8**2+2.95**2-12.62**2 }{ 2 * 9.8 * 2.95 } ) = 160° 31'52" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.62**2+2.95**2-9.8**2 }{ 2 * 12.62 * 2.95 } ) = 15° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 12.62**2+9.8**2-2.95**2 }{ 2 * 12.62 * 9.8 } ) = 4° 28'8" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.82 }{ 12.69 } = 0.38 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.62 }{ 2 * sin 160° 31'52" } = 18.93 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.8**2+2 * 2.95**2 - 12.62**2 } }{ 2 } = 3.543 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.95**2+2 * 12.62**2 - 9.8**2 } }{ 2 } = 7.744 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.8**2+2 * 12.62**2 - 2.95**2 } }{ 2 } = 11.202 ; ;
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