Triangle calculator SSA

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Triangle has two solutions with side c=69.75988335708 and with side c=21.20216159717

#1 Obtuse scalene triangle.

Sides: a = 52   b = 35   c = 69.75988335708

Area: T = 879.314359392
Perimeter: p = 156.7598833571
Semiperimeter: s = 78.37994167854

Angle ∠ A = α = 46.07883111663° = 46°4'42″ = 0.80442182436 rad
Angle ∠ B = β = 29° = 0.50661454831 rad
Angle ∠ C = γ = 104.9221688834° = 104°55'18″ = 1.83112289269 rad

Height: ha = 33.82197536123
Height: hb = 50.24664910811
Height: hc = 25.21101002528

Median: ma = 48.67990245443
Median: mb = 58.98221789237
Median: mc = 27.34882409802

Inradius: r = 11.21986799798
Circumradius: R = 36.09766434435

Vertex coordinates: A[69.75988335708; 0] B[0; 0] C[45.48802247712; 25.21101002528]
Centroid: CG[38.41330194474; 8.40333667509]
Coordinates of the circumscribed circle: U[34.87994167854; -9.29548347266]
Coordinates of the inscribed circle: I[43.37994167854; 11.21986799798]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.9221688834° = 133°55'18″ = 0.80442182436 rad
∠ B' = β' = 151° = 0.50661454831 rad
∠ C' = γ' = 75.07883111663° = 75°4'42″ = 1.83112289269 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 52 ; ; b = 35 ; ; beta = 29° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 35**2 = 52**2 + c**2 -2 * 52 * c * cos (29° ) ; ; ; ; c**2 -90.96c +1479 =0 ; ; p=1; q=-90.96; r=1479 ; ; D = q**2 - 4pr = 90.96**2 - 4 * 1 * 1479 = 2357.80338097 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 90.96 ± sqrt{ 2357.8 } }{ 2 } ; ; c_{1,2} = 45.48022477 ± 24.2786087996 ; ; c_{1} = 69.7588335696 ; ;
c_{2} = 21.2016159704 ; ; ; ; (c -69.7588335696) (c -21.2016159704) = 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 = 52 ; ; b = 35 ; ; c = 69.76 ; ;

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

p = a+b+c = 52+35+69.76 = 156.76 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 156.76 }{ 2 } = 78.38 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 78.38 * (78.38-52)(78.38-35)(78.38-69.76) } ; ; T = sqrt{ 773192.4 } = 879.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 879.31 }{ 52 } = 33.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 879.31 }{ 35 } = 50.25 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 879.31 }{ 69.76 } = 25.21 ; ;

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{ 52**2-35**2-69.76**2 }{ 2 * 35 * 69.76 } ) = 46° 4'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 35**2-52**2-69.76**2 }{ 2 * 52 * 69.76 } ) = 29° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 69.76**2-52**2-35**2 }{ 2 * 35 * 52 } ) = 104° 55'18" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 879.31 }{ 78.38 } = 11.22 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 52 }{ 2 * sin 46° 4'42" } = 36.1 ; ;





#2 Obtuse scalene triangle.

Sides: a = 52   b = 35   c = 21.20216159717

Area: T = 267.2477432084
Perimeter: p = 108.2021615972
Semiperimeter: s = 54.10108079858

Angle ∠ A = α = 133.9221688834° = 133°55'18″ = 2.337737441 rad
Angle ∠ B = β = 29° = 0.50661454831 rad
Angle ∠ C = γ = 17.07883111663° = 17°4'42″ = 0.29880727605 rad

Height: ha = 10.27987473878
Height: hb = 15.27112818333
Height: hc = 25.21101002528

Median: ma = 12.69985928317
Median: mb = 35.64441335973
Median: mc = 43.03662971229

Inradius: r = 4.94398048205
Circumradius: R = 36.09766434435

Vertex coordinates: A[21.20216159717; 0] B[0; 0] C[45.48802247712; 25.21101002528]
Centroid: CG[22.22772802476; 8.40333667509]
Coordinates of the circumscribed circle: U[10.60108079858; 34.50549349794]
Coordinates of the inscribed circle: I[19.10108079858; 4.94398048205]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 46.07883111663° = 46°4'42″ = 2.337737441 rad
∠ B' = β' = 151° = 0.50661454831 rad
∠ C' = γ' = 162.9221688834° = 162°55'18″ = 0.29880727605 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 52 ; ; b = 35 ; ; beta = 29° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 35**2 = 52**2 + c**2 -2 * 52 * c * cos (29° ) ; ; ; ; c**2 -90.96c +1479 =0 ; ; p=1; q=-90.96; r=1479 ; ; D = q**2 - 4pr = 90.96**2 - 4 * 1 * 1479 = 2357.80338097 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 90.96 ± sqrt{ 2357.8 } }{ 2 } ; ; c_{1,2} = 45.48022477 ± 24.2786087996 ; ; c_{1} = 69.7588335696 ; ; : Nr. 1
c_{2} = 21.2016159704 ; ; ; ; (c -69.7588335696) (c -21.2016159704) = 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 = 52 ; ; b = 35 ; ; c = 21.2 ; ;

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

p = a+b+c = 52+35+21.2 = 108.2 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 108.2 }{ 2 } = 54.1 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 54.1 * (54.1-52)(54.1-35)(54.1-21.2) } ; ; T = sqrt{ 71421.19 } = 267.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 267.25 }{ 52 } = 10.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 267.25 }{ 35 } = 15.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 267.25 }{ 21.2 } = 25.21 ; ;

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{ 52**2-35**2-21.2**2 }{ 2 * 35 * 21.2 } ) = 133° 55'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 35**2-52**2-21.2**2 }{ 2 * 52 * 21.2 } ) = 29° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21.2**2-52**2-35**2 }{ 2 * 35 * 52 } ) = 17° 4'42" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 267.25 }{ 54.1 } = 4.94 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 52 }{ 2 * sin 133° 55'18" } = 36.1 ; ;




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