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?

1. Use Law of Cosines

a = 41 ; ; b = 35 ; ; beta = 44° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 35**2 = 41**2 + c**2 -2 * 41 * c * cos (44° ) ; ; ; ; c**2 -58.986c +456 =0 ; ; p=1; q=-58.986; 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} = 29.49293181 ± 20.3428863974 ; ; c_{1} = 49.8358182074 ; ;
c_{2} = 9.15004541258 ; ; ; ; (c -49.8358182074) (c -9.15004541258) = 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 = 49.84 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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**2-49.84**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" ; ;

7. Inradius

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

8. 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 - 2ac cos beta ; ; 35**2 = 41**2 + c**2 -2 * 41 * c * cos (44° ) ; ; ; ; c**2 -58.986c +456 =0 ; ; p=1; q=-58.986; 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} = 29.49293181 ± 20.3428863974 ; ; c_{1} = 49.8358182074 ; ; : Nr. 1
c_{2} = 9.15004541258 ; ; ; ; (c -49.8358182074) (c -9.15004541258) = 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 = 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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