Triangle calculator SSA

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Triangle has two solutions with side c=69.14881516975 and with side c=22.77771814768

#1 Acute scalene triangle.

Sides: a = 60   b = 45   c = 69.14881516975

Area: T = 1333.427725432
Perimeter: p = 174.1488151697
Semiperimeter: s = 87.07440758487

Angle ∠ A = α = 58.98769695348° = 58°59'13″ = 1.03295168342 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 81.01330304652° = 81°47″ = 1.41439441186 rad

Height: ha = 44.44875751438
Height: hb = 59.26334335251
Height: hc = 38.56772565812

Median: ma = 50.03223239675
Median: mb = 60.76999459768
Median: mc = 40.2143595701

Inradius: r = 15.31437112432
Circumradius: R = 35.00437861044

Vertex coordinates: A[69.14881516975; 0] B[0; 0] C[45.96326665871; 38.56772565812]
Centroid: CG[38.37702727615; 12.85657521937]
Coordinates of the circumscribed circle: U[34.57440758487; 5.46879357025]
Coordinates of the inscribed circle: I[42.07440758487; 15.31437112432]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.0133030465° = 121°47″ = 1.03295168342 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 98.98769695348° = 98°59'13″ = 1.41439441186 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 = 60 ; ; b = 45 ; ; c = 69.15 ; ;

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

p = a+b+c = 60+45+69.15 = 174.15 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 174.15 }{ 2 } = 87.07 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 87.07 * (87.07-60)(87.07-45)(87.07-69.15) } ; ; T = sqrt{ 1778028.24 } = 1333.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1333.43 }{ 60 } = 44.45 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1333.43 }{ 45 } = 59.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1333.43 }{ 69.15 } = 38.57 ; ;

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{ 60**2-45**2-69.15**2 }{ 2 * 45 * 69.15 } ) = 58° 59'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 45**2-60**2-69.15**2 }{ 2 * 60 * 69.15 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 69.15**2-60**2-45**2 }{ 2 * 45 * 60 } ) = 81° 47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1333.43 }{ 87.07 } = 15.31 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 60 }{ 2 * sin 58° 59'13" } = 35 ; ;





#2 Obtuse scalene triangle.

Sides: a = 60   b = 45   c = 22.77771814768

Area: T = 439.2276701107
Perimeter: p = 127.7777181477
Semiperimeter: s = 63.88985907384

Angle ∠ A = α = 121.0133030465° = 121°47″ = 2.11220758194 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 18.98769695348° = 18°59'13″ = 0.33113851334 rad

Height: ha = 14.64108900369
Height: hb = 19.52111867158
Height: hc = 38.56772565812

Median: ma = 19.2854708917
Median: mb = 39.41100240804
Median: mc = 51.79657527312

Inradius: r = 6.8754884796
Circumradius: R = 35.00437861044

Vertex coordinates: A[22.77771814768; 0] B[0; 0] C[45.96326665871; 38.56772565812]
Centroid: CG[22.9133282688; 12.85657521937]
Coordinates of the circumscribed circle: U[11.38985907384; 33.09993208787]
Coordinates of the inscribed circle: I[18.88985907384; 6.8754884796]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 58.98769695348° = 58°59'13″ = 2.11220758194 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 161.0133030465° = 161°47″ = 0.33113851334 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 60 ; ; b = 45 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 45**2 = 60**2 + c**2 -2 * 45 * c * cos (40° ) ; ; ; ; c**2 -91.925c +1575 =0 ; ; p=1; q=-91.9253331743; r=1575 ; ; D = q**2 - 4pr = 91.925**2 - 4 * 1 * 1575 = 2150.2668792 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 91.93 ± sqrt{ 2150.27 } }{ 2 } ; ; c_{1,2} = 45.9626665871 ± 23.1854851103 ; ; c_{1} = 69.1481516975 ; ;
c_{2} = 22.7771814768 ; ; ; ; (c -69.1481516975) (c -22.7771814768) = 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 = 60 ; ; b = 45 ; ; c = 22.78 ; ;

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

p = a+b+c = 60+45+22.78 = 127.78 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 127.78 }{ 2 } = 63.89 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 63.89 * (63.89-60)(63.89-45)(63.89-22.78) } ; ; T = sqrt{ 192920.09 } = 439.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 439.23 }{ 60 } = 14.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 439.23 }{ 45 } = 19.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 439.23 }{ 22.78 } = 38.57 ; ;

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{ 60**2-45**2-22.78**2 }{ 2 * 45 * 22.78 } ) = 121° 47" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 45**2-60**2-22.78**2 }{ 2 * 60 * 22.78 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22.78**2-60**2-45**2 }{ 2 * 45 * 60 } ) = 18° 59'13" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 439.23 }{ 63.89 } = 6.87 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 60 }{ 2 * sin 121° 47" } = 35 ; ;




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