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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 35**2 = 52**2 + c**2 -2 * 35 * c * cos (29° ) ; ; ; ; c**2 -90.96c +1479 =0 ; ; p=1; q=-90.9604495425; 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} = 45.4802247712 ± 24.2786087996 ; ; c_{1} = 69.7588335708 ; ;
c_{2} = 21.2016159717 ; ; ; ; (c -69.7588335708) (c -21.2016159717) = 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 = 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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