Triangle calculator SSA

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Triangle has two solutions with side c=20.35108001321 and with side c=9.41109321873

#1 Acute scalene triangle.

Sides: a = 24.9   b = 20.7   c = 20.35108001321

Area: T = 203.1443859791
Perimeter: p = 65.95108001321
Semiperimeter: s = 32.97554000661

Angle ∠ A = α = 74.67877252987° = 74°40'40″ = 1.30333721844 rad
Angle ∠ B = β = 53.3° = 53°18' = 0.93302604913 rad
Angle ∠ C = γ = 52.02222747013° = 52°1'20″ = 0.90879599779 rad

Height: ha = 16.31767758868
Height: hb = 19.62774260667
Height: hc = 19.96442135417

Median: ma = 16.31993147224
Median: mb = 20.24774697928
Median: mc = 20.51112465125

Inradius: r = 6.16604668748
Circumradius: R = 12.90988480977

Vertex coordinates: A[20.35108001321; 0] B[0; 0] C[14.88108661597; 19.96442135417]
Centroid: CG[11.74438887639; 6.65547378472]
Coordinates of the circumscribed circle: U[10.17554000661; 7.94435252065]
Coordinates of the inscribed circle: I[12.27554000661; 6.16604668748]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105.3222274701° = 105°19'20″ = 1.30333721844 rad
∠ B' = β' = 126.7° = 126°42' = 0.93302604913 rad
∠ C' = γ' = 127.9787725299° = 127°58'40″ = 0.90879599779 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 24.9 ; ; b = 20.7 ; ; beta = 53° 18' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 20.7**2 = 24.9**2 + c**2 -2 * 24.9 * c * cos (53° 18') ; ; ; ; c**2 -29.762c +191.52 =0 ; ; p=1; q=-29.762; r=191.52 ; ; D = q**2 - 4pr = 29.762**2 - 4 * 1 * 191.52 = 119.68071065 ; ; D>0 ; ;
 ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 29.76 ± sqrt{ 119.68 } }{ 2 } ; ; c_{1,2} = 14.88086616 ± 5.46993397242 ; ; c_{1} = 20.3508001321 ; ; c_{2} = 9.41093218727 ; ; ; ; text{ Factored form: } ; ; (c -20.3508001321) (c -9.41093218727) = 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 = 24.9 ; ; b = 20.7 ; ; c = 20.35 ; ;

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

p = a+b+c = 24.9+20.7+20.35 = 65.95 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 65.95 }{ 2 } = 32.98 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.98 * (32.98-24.9)(32.98-20.7)(32.98-20.35) } ; ; T = sqrt{ 41267.43 } = 203.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 203.14 }{ 24.9 } = 16.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 203.14 }{ 20.7 } = 19.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 203.14 }{ 20.35 } = 19.96 ; ;

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{ 20.7**2+20.35**2-24.9**2 }{ 2 * 20.7 * 20.35 } ) = 74° 40'40" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 24.9**2+20.35**2-20.7**2 }{ 2 * 24.9 * 20.35 } ) = 53° 18' ; ;
 gamma = 180° - alpha - beta = 180° - 74° 40'40" - 53° 18' = 52° 1'20" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 203.14 }{ 32.98 } = 6.16 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 24.9 }{ 2 * sin 74° 40'40" } = 12.91 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.7**2+2 * 20.35**2 - 24.9**2 } }{ 2 } = 16.319 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.35**2+2 * 24.9**2 - 20.7**2 } }{ 2 } = 20.247 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.7**2+2 * 24.9**2 - 20.35**2 } }{ 2 } = 20.511 ; ;



#2 Obtuse scalene triangle.

Sides: a = 24.9   b = 20.7   c = 9.41109321873

Area: T = 93.94109299064
Perimeter: p = 55.01109321873
Semiperimeter: s = 27.50554660936

Angle ∠ A = α = 105.3222274701° = 105°19'20″ = 1.83882204692 rad
Angle ∠ B = β = 53.3° = 53°18' = 0.93302604913 rad
Angle ∠ C = γ = 21.37877252987° = 21°22'40″ = 0.3733111693 rad

Height: ha = 7.54554562174
Height: hb = 9.07664183484
Height: hc = 19.96442135417

Median: ma = 10.17547394225
Median: mb = 15.72114923693
Median: mc = 22.408777965

Inradius: r = 3.41553549548
Circumradius: R = 12.90988480977

Vertex coordinates: A[9.41109321873; 0] B[0; 0] C[14.88108661597; 19.96442135417]
Centroid: CG[8.09772661157; 6.65547378472]
Coordinates of the circumscribed circle: U[4.70554660936; 12.02106883352]
Coordinates of the inscribed circle: I[6.80554660936; 3.41553549548]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 74.67877252987° = 74°40'40″ = 1.83882204692 rad
∠ B' = β' = 126.7° = 126°42' = 0.93302604913 rad
∠ C' = γ' = 158.6222274701° = 158°37'20″ = 0.3733111693 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 24.9 ; ; b = 20.7 ; ; beta = 53° 18' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 20.7**2 = 24.9**2 + c**2 -2 * 24.9 * c * cos (53° 18') ; ; ; ; c**2 -29.762c +191.52 =0 ; ; p=1; q=-29.762; r=191.52 ; ; D = q**2 - 4pr = 29.762**2 - 4 * 1 * 191.52 = 119.68071065 ; ; D>0 ; ; : Nr. 1
 ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 29.76 ± sqrt{ 119.68 } }{ 2 } ; ; c_{1,2} = 14.88086616 ± 5.46993397242 ; ; c_{1} = 20.3508001321 ; ; c_{2} = 9.41093218727 ; ; ; ; text{ Factored form: } ; ; (c -20.3508001321) (c -9.41093218727) = 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 = 24.9 ; ; b = 20.7 ; ; c = 9.41 ; ;

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

p = a+b+c = 24.9+20.7+9.41 = 55.01 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55.01 }{ 2 } = 27.51 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.51 * (27.51-24.9)(27.51-20.7)(27.51-9.41) } ; ; T = sqrt{ 8824.9 } = 93.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 93.94 }{ 24.9 } = 7.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 93.94 }{ 20.7 } = 9.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 93.94 }{ 9.41 } = 19.96 ; ;

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{ 20.7**2+9.41**2-24.9**2 }{ 2 * 20.7 * 9.41 } ) = 105° 19'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 24.9**2+9.41**2-20.7**2 }{ 2 * 24.9 * 9.41 } ) = 53° 18' ; ;
 gamma = 180° - alpha - beta = 180° - 105° 19'20" - 53° 18' = 21° 22'40" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 93.94 }{ 27.51 } = 3.42 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 24.9 }{ 2 * sin 105° 19'20" } = 12.91 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.7**2+2 * 9.41**2 - 24.9**2 } }{ 2 } = 10.175 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.41**2+2 * 24.9**2 - 20.7**2 } }{ 2 } = 15.721 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.7**2+2 * 24.9**2 - 9.41**2 } }{ 2 } = 22.408 ; ;
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