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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 20.7**2 = 24.9**2 + c**2 -2 * 20.7 * c * cos (53° 18') ; ; ; ; c**2 -29.762c +191.52 =0 ; ; p=1; q=-29.7617323194; 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} = 14.8808661597 ± 5.46993397242 ; ;
c_{1} = 20.3508001321 ; ; c_{2} = 9.41093218727 ; ; ; ; (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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 24.9**2-20.7**2-9.41**2 }{ 2 * 20.7 * 9.41 } ) = 105° 19'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20.7**2-24.9**2-9.41**2 }{ 2 * 24.9 * 9.41 } ) = 53° 18' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.41**2-24.9**2-20.7**2 }{ 2 * 20.7 * 24.9 } ) = 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 ; ;




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