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?

1. Use Law of Cosines

a = 115 ; ; b = 90 ; ; beta = 50° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 115**2 + c**2 -2 * 115 * c * cos (50° ) ; ; ; ; c**2 -147.841c +5125 =0 ; ; p=1; q=-147.841; 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} = 73.92057511 ± 18.4187791446 ; ; c_{1} = 92.3393542585 ; ; c_{2} = 55.5017959694 ; ; ; ; text{ Factored form: } ; ; (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 = 92.34 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 90**2+92.34**2-115**2 }{ 2 * 90 * 92.34 } ) = 78° 11'27" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 115**2+92.34**2-90**2 }{ 2 * 115 * 92.34 } ) = 50° ; ;
 gamma = 180° - alpha - beta = 180° - 78° 11'27" - 50° = 51° 48'33" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 92.34**2 - 115**2 } }{ 2 } = 70.76 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 92.34**2+2 * 115**2 - 90**2 } }{ 2 } = 94.079 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 115**2 - 92.34**2 } }{ 2 } = 92.363 ; ;



#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 115 ; ; b = 90 ; ; beta = 50° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 115**2 + c**2 -2 * 115 * c * cos (50° ) ; ; ; ; c**2 -147.841c +5125 =0 ; ; p=1; q=-147.841; 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 } ; ; : Nr. 1
c_{1,2} = 73.92057511 ± 18.4187791446 ; ; c_{1} = 92.3393542585 ; ; c_{2} = 55.5017959694 ; ; ; ; text{ Factored form: } ; ; (c -92.3393542585) (c -55.5017959694) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 90**2+55.5**2-115**2 }{ 2 * 90 * 55.5 } ) = 101° 48'33" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 115**2+55.5**2-90**2 }{ 2 * 115 * 55.5 } ) = 50° ; ;
 gamma = 180° - alpha - beta = 180° - 101° 48'33" - 50° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 55.5**2 - 115**2 } }{ 2 } = 47.791 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 55.5**2+2 * 115**2 - 90**2 } }{ 2 } = 78.28 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 115**2 - 55.5**2 } }{ 2 } = 99.46 ; ;
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