Triangle calculator SSA

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Triangle has two solutions with side c=49.83658182113 and with side c=9.15500454165

#1 Acute scalene triangle.

Sides: a = 41   b = 35   c = 49.83658182113

Area: T = 709.6876799518
Perimeter: p = 125.8365818211
Semiperimeter: s = 62.91879091057

Angle ∠ A = α = 54.46332282571° = 54°27'48″ = 0.95105626544 rad
Angle ∠ B = β = 44° = 0.76879448709 rad
Angle ∠ C = γ = 81.53767717429° = 81°32'12″ = 1.42330851284 rad

Height: ha = 34.61988682692
Height: hb = 40.5543531401
Height: hc = 28.48109931888

Median: ma = 37.86989105784
Median: mb = 42.14332602962
Median: mc = 28.84661055569

Inradius: r = 11.28795674492
Circumradius: R = 25.19222394435

Vertex coordinates: A[49.83658182113; 0] B[0; 0] C[29.49329318139; 28.48109931888]
Centroid: CG[26.44329166751; 9.49436643963]
Coordinates of the circumscribed circle: U[24.91879091057; 3.70876588271]
Coordinates of the inscribed circle: I[27.91879091057; 11.28795674492]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.5376771743° = 125°32'12″ = 0.95105626544 rad
∠ B' = β' = 136° = 0.76879448709 rad
∠ C' = γ' = 98.46332282571° = 98°27'48″ = 1.42330851284 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 = 41 ; ; b = 35 ; ; c = 49.84 ; ;

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

p = a+b+c = 41+35+49.84 = 125.84 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 125.84 }{ 2 } = 62.92 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 62.92 * (62.92-41)(62.92-35)(62.92-49.84) } ; ; T = sqrt{ 503655.35 } = 709.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 709.69 }{ 41 } = 34.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 709.69 }{ 35 } = 40.55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 709.69 }{ 49.84 } = 28.48 ; ;

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{ 41**2-35**2-49.84**2 }{ 2 * 35 * 49.84 } ) = 54° 27'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 35**2-41**2-49.84**2 }{ 2 * 41 * 49.84 } ) = 44° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 49.84**2-41**2-35**2 }{ 2 * 35 * 41 } ) = 81° 32'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 709.69 }{ 62.92 } = 11.28 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 41 }{ 2 * sin 54° 27'48" } = 25.19 ; ;





#2 Obtuse scalene triangle.

Sides: a = 41   b = 35   c = 9.15500454165

Area: T = 130.3011190592
Perimeter: p = 85.15500454165
Semiperimeter: s = 42.57550227082

Angle ∠ A = α = 125.5376771743° = 125°32'12″ = 2.19110299992 rad
Angle ∠ B = β = 44° = 0.76879448709 rad
Angle ∠ C = γ = 10.46332282571° = 10°27'48″ = 0.18326177835 rad

Height: ha = 6.35661556386
Height: hb = 7.44657823195
Height: hc = 28.48109931888

Median: ma = 15.30107080085
Median: mb = 24.00223262531
Median: mc = 37.84326897461

Inradius: r = 3.06105078354
Circumradius: R = 25.19222394435

Vertex coordinates: A[9.15500454165; 0] B[0; 0] C[29.49329318139; 28.48109931888]
Centroid: CG[12.88109924101; 9.49436643963]
Coordinates of the circumscribed circle: U[4.57550227082; 24.77333343617]
Coordinates of the inscribed circle: I[7.57550227082; 3.06105078354]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 54.46332282571° = 54°27'48″ = 2.19110299992 rad
∠ B' = β' = 136° = 0.76879448709 rad
∠ C' = γ' = 169.5376771743° = 169°32'12″ = 0.18326177835 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 41 ; ; b = 35 ; ; beta = 44° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 35**2 = 41**2 + c**2 -2 * 35 * c * cos (44° ) ; ; ; ; c**2 -58.986c +456 =0 ; ; p=1; q=-58.9858636278; r=456 ; ; D = q**2 - 4pr = 58.986**2 - 4 * 1 * 456 = 1655.33210791 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 58.99 ± sqrt{ 1655.33 } }{ 2 } ; ; c_{1,2} = 29.4929318139 ± 20.3428863974 ; ; c_{1} = 49.8358182113 ; ;
c_{2} = 9.15004541646 ; ; ; ; (c -49.8358182113) (c -9.15004541646) = 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 = 41 ; ; b = 35 ; ; c = 9.15 ; ;

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

p = a+b+c = 41+35+9.15 = 85.15 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 85.15 }{ 2 } = 42.58 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42.58 * (42.58-41)(42.58-35)(42.58-9.15) } ; ; T = sqrt{ 16978.4 } = 130.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 130.3 }{ 41 } = 6.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 130.3 }{ 35 } = 7.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 130.3 }{ 9.15 } = 28.48 ; ;

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{ 41**2-35**2-9.15**2 }{ 2 * 35 * 9.15 } ) = 125° 32'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 35**2-41**2-9.15**2 }{ 2 * 41 * 9.15 } ) = 44° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.15**2-41**2-35**2 }{ 2 * 35 * 41 } ) = 10° 27'48" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 130.3 }{ 42.58 } = 3.06 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 41 }{ 2 * sin 125° 32'12" } = 25.19 ; ;




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