Triangle calculator SSA

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Triangle has two solutions with side c=42.39993927704 and with side c=9.3166171157

#1 Acute scalene triangle.

Sides: a = 42   b = 37   c = 42.39993927704

Area: T = 701.6354726439
Perimeter: p = 121.399939277
Semiperimeter: s = 60.76996963852

Angle ∠ A = α = 63.44441109646° = 63°26'39″ = 1.10773086273 rad
Angle ∠ B = β = 52° = 0.9087571211 rad
Angle ∠ C = γ = 64.55658890354° = 64°33'21″ = 1.12767128152 rad

Height: ha = 33.41111774495
Height: hb = 37.92662014291
Height: hc = 33.09664516515

Median: ma = 33.79987315391
Median: mb = 37.92989368906
Median: mc = 33.42326401287

Inradius: r = 11.55991142662
Circumradius: R = 23.47768369788

Vertex coordinates: A[42.39993927704; 0] B[0; 0] C[25.85877819637; 33.09664516515]
Centroid: CG[22.7522391578; 11.03221505505]
Coordinates of the circumscribed circle: U[21.21996963852; 10.08663644445]
Coordinates of the inscribed circle: I[23.76996963852; 11.55991142662]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.5565889035° = 116°33'21″ = 1.10773086273 rad
∠ B' = β' = 128° = 0.9087571211 rad
∠ C' = γ' = 115.4444110965° = 115°26'39″ = 1.12767128152 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 42 ; ; b = 37 ; ; beta = 52° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 37**2 = 42**2 + c**2 -2 * 42 * c * cos (52° ) ; ; ; ; c**2 -51.716c +395 =0 ; ; p=1; q=-51.716; r=395 ; ; D = q**2 - 4pr = 51.716**2 - 4 * 1 * 395 = 1094.49955232 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 51.72 ± sqrt{ 1094.5 } }{ 2 } ; ; c_{1,2} = 25.85778196 ± 16.5416108067 ; ; c_{1} = 42.3993927667 ; ;
c_{2} = 9.31617115328 ; ; ; ; (c -42.3993927667) (c -9.31617115328) = 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 = 42 ; ; b = 37 ; ; c = 42.4 ; ;

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

p = a+b+c = 42+37+42.4 = 121.4 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 121.4 }{ 2 } = 60.7 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 60.7 * (60.7-42)(60.7-37)(60.7-42.4) } ; ; T = sqrt{ 492291.29 } = 701.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 701.63 }{ 42 } = 33.41 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 701.63 }{ 37 } = 37.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 701.63 }{ 42.4 } = 33.1 ; ;

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{ 42**2-37**2-42.4**2 }{ 2 * 37 * 42.4 } ) = 63° 26'39" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 37**2-42**2-42.4**2 }{ 2 * 42 * 42.4 } ) = 52° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 42.4**2-42**2-37**2 }{ 2 * 37 * 42 } ) = 64° 33'21" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 701.63 }{ 60.7 } = 11.56 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 42 }{ 2 * sin 63° 26'39" } = 23.48 ; ;





#2 Obtuse scalene triangle.

Sides: a = 42   b = 37   c = 9.3166171157

Area: T = 154.1666104136
Perimeter: p = 88.3166171157
Semiperimeter: s = 44.15880855785

Angle ∠ A = α = 116.5565889035° = 116°33'21″ = 2.03442840263 rad
Angle ∠ B = β = 52° = 0.9087571211 rad
Angle ∠ C = γ = 11.44441109646° = 11°26'39″ = 0.21997374163 rad

Height: ha = 7.34112430541
Height: hb = 8.33333029263
Height: hc = 33.09664516515

Median: ma = 16.93879905099
Median: mb = 24.14884062106
Median: mc = 39.30439723023

Inradius: r = 3.49112316084
Circumradius: R = 23.47768369788

Vertex coordinates: A[9.3166171157; 0] B[0; 0] C[25.85877819637; 33.09664516515]
Centroid: CG[11.72546510402; 11.03221505505]
Coordinates of the circumscribed circle: U[4.65880855785; 23.0110087207]
Coordinates of the inscribed circle: I[7.15880855785; 3.49112316084]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 63.44441109646° = 63°26'39″ = 2.03442840263 rad
∠ B' = β' = 128° = 0.9087571211 rad
∠ C' = γ' = 168.5565889035° = 168°33'21″ = 0.21997374163 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 42 ; ; b = 37 ; ; beta = 52° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 37**2 = 42**2 + c**2 -2 * 42 * c * cos (52° ) ; ; ; ; c**2 -51.716c +395 =0 ; ; p=1; q=-51.716; r=395 ; ; D = q**2 - 4pr = 51.716**2 - 4 * 1 * 395 = 1094.49955232 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 51.72 ± sqrt{ 1094.5 } }{ 2 } ; ; c_{1,2} = 25.85778196 ± 16.5416108067 ; ; c_{1} = 42.3993927667 ; ; : Nr. 1
c_{2} = 9.31617115328 ; ; ; ; (c -42.3993927667) (c -9.31617115328) = 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 = 42 ; ; b = 37 ; ; c = 9.32 ; ;

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

p = a+b+c = 42+37+9.32 = 88.32 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 88.32 }{ 2 } = 44.16 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 44.16 * (44.16-42)(44.16-37)(44.16-9.32) } ; ; T = sqrt{ 23767.19 } = 154.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 154.17 }{ 42 } = 7.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 154.17 }{ 37 } = 8.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 154.17 }{ 9.32 } = 33.1 ; ;

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{ 42**2-37**2-9.32**2 }{ 2 * 37 * 9.32 } ) = 116° 33'21" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 37**2-42**2-9.32**2 }{ 2 * 42 * 9.32 } ) = 52° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.32**2-42**2-37**2 }{ 2 * 37 * 42 } ) = 11° 26'39" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 154.17 }{ 44.16 } = 3.49 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 42 }{ 2 * sin 116° 33'21" } = 23.48 ; ;




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