Triangle calculator SSA

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Triangle has two solutions with side c=92.33993542585 and with side c=55.50217959694

#1 Acute scalene triangle.

Sides: a = 115   b = 90   c = 92.33993542585

Area: T = 4067.323282963
Perimeter: p = 297.3399354259
Semiperimeter: s = 148.6769677129

Angle ∠ A = α = 78.19108072946° = 78°11'27″ = 1.36546870321 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 51.80991927054° = 51°48'33″ = 0.90442409955 rad

Height: ha = 70.7366049211
Height: hb = 90.38549517696
Height: hc = 88.09551109587

Median: ma = 70.7660357351
Median: mb = 94.07985744601
Median: mc = 92.36326597375

Inradius: r = 27.35881197469
Circumradius: R = 58.743332802

Vertex coordinates: A[92.33993542585; 0] B[0; 0] C[73.9210575114; 88.09551109587]
Centroid: CG[55.42199764575; 29.36550369862]
Coordinates of the circumscribed circle: U[46.17696771293; 36.32199600859]
Coordinates of the inscribed circle: I[58.67696771293; 27.35881197469]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 101.8099192705° = 101°48'33″ = 1.36546870321 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 128.1910807295° = 128°11'27″ = 0.90442409955 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 = 115 ; ; b = 90 ; ; c = 92.34 ; ;

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

p = a+b+c = 115+90+92.34 = 297.34 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 297.34 }{ 2 } = 148.67 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 148.67 * (148.67-115)(148.67-90)(148.67-92.34) } ; ; T = sqrt{ 16543115 } = 4067.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4067.32 }{ 115 } = 70.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4067.32 }{ 90 } = 90.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4067.32 }{ 92.34 } = 88.1 ; ;

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{ 115**2-90**2-92.34**2 }{ 2 * 90 * 92.34 } ) = 78° 11'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-115**2-92.34**2 }{ 2 * 115 * 92.34 } ) = 50° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 92.34**2-115**2-90**2 }{ 2 * 90 * 115 } ) = 51° 48'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4067.32 }{ 148.67 } = 27.36 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 115 }{ 2 * sin 78° 11'27" } = 58.74 ; ;





#2 Obtuse scalene triangle.

Sides: a = 115   b = 90   c = 55.50217959694

Area: T = 2444.718843716
Perimeter: p = 260.5021795969
Semiperimeter: s = 130.2510897985

Angle ∠ A = α = 101.8099192705° = 101°48'33″ = 1.77769056215 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 28.19108072946° = 28°11'27″ = 0.49220224061 rad

Height: ha = 42.51768423854
Height: hb = 54.32770763814
Height: hc = 88.09551109587

Median: ma = 47.79109476566
Median: mb = 78.28797846057
Median: mc = 99.46604829118

Inradius: r = 18.76993019779
Circumradius: R = 58.743332802

Vertex coordinates: A[55.50217959694; 0] B[0; 0] C[73.9210575114; 88.09551109587]
Centroid: CG[43.14107903611; 29.36550369862]
Coordinates of the circumscribed circle: U[27.75108979847; 51.77551508728]
Coordinates of the inscribed circle: I[40.25108979847; 18.76993019779]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 78.19108072946° = 78°11'27″ = 1.77769056215 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 151.8099192705° = 151°48'33″ = 0.49220224061 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 115 ; ; b = 90 ; ; beta = 50° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 115**2 + c**2 -2 * 90 * c * cos (50° ) ; ; ; ; c**2 -147.841c +5125 =0 ; ; p=1; q=-147.841150228; r=5125 ; ; D = q**2 - 4pr = 147.841**2 - 4 * 1 * 5125 = 1357.00570071 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 147.84 ± sqrt{ 1357.01 } }{ 2 } ; ; c_{1,2} = 73.920575114 ± 18.4187791446 ; ; c_{1} = 92.3393542585 ; ;
c_{2} = 55.5017959694 ; ; ; ; (c -92.3393542585) (c -55.5017959694) = 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 = 115 ; ; b = 90 ; ; c = 55.5 ; ;

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

p = a+b+c = 115+90+55.5 = 260.5 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 260.5 }{ 2 } = 130.25 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 130.25 * (130.25-115)(130.25-90)(130.25-55.5) } ; ; T = sqrt{ 5976648.24 } = 2444.72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2444.72 }{ 115 } = 42.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2444.72 }{ 90 } = 54.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2444.72 }{ 55.5 } = 88.1 ; ;

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{ 115**2-90**2-55.5**2 }{ 2 * 90 * 55.5 } ) = 101° 48'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-115**2-55.5**2 }{ 2 * 115 * 55.5 } ) = 50° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 55.5**2-115**2-90**2 }{ 2 * 90 * 115 } ) = 28° 11'27" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2444.72 }{ 130.25 } = 18.77 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 115 }{ 2 * sin 101° 48'33" } = 58.74 ; ;




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