Triangle calculator SSA

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Triangle has two solutions with side c=488.0166237676 and with side c=289.1329723781

#1 Acute scalene triangle.

Sides: a = 500   b = 330   c = 488.0166237676

Area: T = 76779.64223831
Perimeter: p = 1318.016623768
Semiperimeter: s = 659.0088118838

Angle ∠ A = α = 72.462170313° = 72°27'42″ = 1.26546953012 rad
Angle ∠ B = β = 39° = 0.68106784083 rad
Angle ∠ C = γ = 68.538829687° = 68°32'18″ = 1.19662189441 rad

Height: ha = 307.1198569533
Height: hb = 465.3311165958
Height: hc = 314.6660195525

Median: ma = 333.2121530589
Median: mb = 465.6776845159
Median: mc = 346.2880288121

Inradius: r = 116.5087885394
Circumradius: R = 262.1887595296

Vertex coordinates: A[488.0166237676; 0] B[0; 0] C[388.5732980728; 314.6660195525]
Centroid: CG[292.1966406135; 104.8876731842]
Coordinates of the circumscribed circle: U[244.0088118838; 95.92990001423]
Coordinates of the inscribed circle: I[329.0088118838; 116.5087885394]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.538829687° = 107°32'18″ = 1.26546953012 rad
∠ B' = β' = 141° = 0.68106784083 rad
∠ C' = γ' = 111.462170313° = 111°27'42″ = 1.19662189441 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 500 ; ; b = 330 ; ; beta = 39° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 330**2 = 500**2 + c**2 -2 * 500 * c * cos (39° ) ; ; ; ; c**2 -777.146c +141100 =0 ; ; p=1; q=-777.146; r=141100 ; ; D = q**2 - 4pr = 777.146**2 - 4 * 1 * 141100 = 39555.8454089 ; ; D>0 ; ; ; ;
c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 777.15 ± sqrt{ 39555.85 } }{ 2 } ; ; c_{1,2} = 388.57298073 ± 99.443256947 ; ; c_{1} = 488.016237675 ; ; c_{2} = 289.129723782 ; ; ; ; text{ Factored form: } ; ; (c -488.016237675) (c -289.129723782) = 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 = 500 ; ; b = 330 ; ; c = 488.02 ; ;

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

p = a+b+c = 500+330+488.02 = 1318.02 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1318.02 }{ 2 } = 659.01 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 659.01 * (659.01-500)(659.01-330)(659.01-488.02) } ; ; T = sqrt{ 5895113484.48 } = 76779.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 76779.64 }{ 500 } = 307.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 76779.64 }{ 330 } = 465.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 76779.64 }{ 488.02 } = 314.66 ; ;

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{ 330**2+488.02**2-500**2 }{ 2 * 330 * 488.02 } ) = 72° 27'42" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 500**2+488.02**2-330**2 }{ 2 * 500 * 488.02 } ) = 39° ; ;
 gamma = 180° - alpha - beta = 180° - 72° 27'42" - 39° = 68° 32'18" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 76779.64 }{ 659.01 } = 116.51 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 500 }{ 2 * sin 72° 27'42" } = 262.19 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 330**2+2 * 488.02**2 - 500**2 } }{ 2 } = 333.212 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 488.02**2+2 * 500**2 - 330**2 } }{ 2 } = 465.677 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 330**2+2 * 500**2 - 488.02**2 } }{ 2 } = 346.28 ; ;



#2 Obtuse scalene triangle.

Sides: a = 500   b = 330   c = 289.1329723781

Area: T = 45488.80877086
Perimeter: p = 1119.132972378
Semiperimeter: s = 559.5654861891

Angle ∠ A = α = 107.538829687° = 107°32'18″ = 1.87768973524 rad
Angle ∠ B = β = 39° = 0.68106784083 rad
Angle ∠ C = γ = 33.462170313° = 33°27'42″ = 0.58440168929 rad

Height: ha = 181.9555230834
Height: hb = 275.6989743688
Height: hc = 314.6660195525

Median: ma = 183.7066283472
Median: mb = 373.5954698285
Median: mc = 398.1854631429

Inradius: r = 81.2933181196
Circumradius: R = 262.1887595296

Vertex coordinates: A[289.1329723781; 0] B[0; 0] C[388.5732980728; 314.6660195525]
Centroid: CG[225.9010901503; 104.8876731842]
Coordinates of the circumscribed circle: U[144.5654861891; 218.7311195383]
Coordinates of the inscribed circle: I[229.5654861891; 81.2933181196]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 72.462170313° = 72°27'42″ = 1.87768973524 rad
∠ B' = β' = 141° = 0.68106784083 rad
∠ C' = γ' = 146.538829687° = 146°32'18″ = 0.58440168929 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 500 ; ; b = 330 ; ; beta = 39° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 330**2 = 500**2 + c**2 -2 * 500 * c * cos (39° ) ; ; ; ; c**2 -777.146c +141100 =0 ; ; p=1; q=-777.146; r=141100 ; ; D = q**2 - 4pr = 777.146**2 - 4 * 1 * 141100 = 39555.8454089 ; ; D>0 ; ; ; ; : Nr. 1
c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 777.15 ± sqrt{ 39555.85 } }{ 2 } ; ; c_{1,2} = 388.57298073 ± 99.443256947 ; ; c_{1} = 488.016237675 ; ; c_{2} = 289.129723782 ; ; ; ; text{ Factored form: } ; ; (c -488.016237675) (c -289.129723782) = 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 = 500 ; ; b = 330 ; ; c = 289.13 ; ;

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

p = a+b+c = 500+330+289.13 = 1119.13 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1119.13 }{ 2 } = 559.56 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 559.56 * (559.56-500)(559.56-330)(559.56-289.13) } ; ; T = sqrt{ 2069231626.75 } = 45488.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 45488.81 }{ 500 } = 181.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 45488.81 }{ 330 } = 275.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 45488.81 }{ 289.13 } = 314.66 ; ;

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{ 330**2+289.13**2-500**2 }{ 2 * 330 * 289.13 } ) = 107° 32'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 500**2+289.13**2-330**2 }{ 2 * 500 * 289.13 } ) = 39° ; ;
 gamma = 180° - alpha - beta = 180° - 107° 32'18" - 39° = 33° 27'42" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 45488.81 }{ 559.56 } = 81.29 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 500 }{ 2 * sin 107° 32'18" } = 262.19 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 330**2+2 * 289.13**2 - 500**2 } }{ 2 } = 183.706 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 289.13**2+2 * 500**2 - 330**2 } }{ 2 } = 373.595 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 330**2+2 * 500**2 - 289.13**2 } }{ 2 } = 398.185 ; ;
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