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?

1. Use Law of Cosines

a = 60 ; ; b = 45 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 45**2 = 60**2 + c**2 -2 * 60 * c * cos (40° ) ; ; ; ; c**2 -91.925c +1575 =0 ; ; p=1; q=-91.925; 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} = 45.96266659 ± 23.1854851103 ; ; c_{1} = 69.1481517003 ; ; c_{2} = 22.7771814797 ; ; ; ; text{ Factored form: } ; ; (c -69.1481517003) (c -22.7771814797) = 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 = 69.15 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 45**2+69.15**2-60**2 }{ 2 * 45 * 69.15 } ) = 58° 59'13" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 60**2+69.15**2-45**2 }{ 2 * 60 * 69.15 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 58° 59'13" - 40° = 81° 47" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 69.15**2 - 60**2 } }{ 2 } = 50.032 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 69.15**2+2 * 60**2 - 45**2 } }{ 2 } = 60.7 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 60**2 - 69.15**2 } }{ 2 } = 40.214 ; ;







#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 60 ; ; b = 45 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 45**2 = 60**2 + c**2 -2 * 60 * c * cos (40° ) ; ; ; ; c**2 -91.925c +1575 =0 ; ; p=1; q=-91.925; 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} = 45.96266659 ± 23.1854851103 ; ; c_{1} = 69.1481517003 ; ; c_{2} = 22.7771814797 ; ; ; ; text{ Factored form: } ; ; (c -69.1481517003) (c -22.7771814797) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 45**2+22.78**2-60**2 }{ 2 * 45 * 22.78 } ) = 121° 47" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 60**2+22.78**2-45**2 }{ 2 * 60 * 22.78 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 121° 47" - 40° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 22.78**2 - 60**2 } }{ 2 } = 19.285 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.78**2+2 * 60**2 - 45**2 } }{ 2 } = 39.41 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 60**2 - 22.78**2 } }{ 2 } = 51.796 ; ;
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