Triangle calculator SSA

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Triangle has two solutions with side c=26.47875894296 and with side c=2.45549062585

#1 Acute scalene triangle.

Sides: a = 33   b = 32   c = 26.47875894296

Area: T = 392.6655345705
Perimeter: p = 91.47875894296
Semiperimeter: s = 45.73987947148

Angle ∠ A = α = 67.95437798724° = 67°57'14″ = 1.18660171979 rad
Angle ∠ B = β = 64° = 1.11770107213 rad
Angle ∠ C = γ = 48.04662201276° = 48°2'46″ = 0.83985647344 rad

Height: ha = 23.79878997397
Height: hb = 24.54215841065
Height: hc = 29.66602035279

Median: ma = 24.2965706843
Median: mb = 25.27990698207
Median: mc = 29.68655910249

Inradius: r = 8.58549517495
Circumradius: R = 17.80216310476

Vertex coordinates: A[26.47875894296; 0] B[0; 0] C[14.4666247844; 29.66602035279]
Centroid: CG[13.64879457579; 9.88767345093]
Coordinates of the circumscribed circle: U[13.23987947148; 11.90109404021]
Coordinates of the inscribed circle: I[13.73987947148; 8.58549517495]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.0466220128° = 112°2'46″ = 1.18660171979 rad
∠ B' = β' = 116° = 1.11770107213 rad
∠ C' = γ' = 131.9543779872° = 131°57'14″ = 0.83985647344 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 = 33 ; ; b = 32 ; ; c = 26.48 ; ;

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

p = a+b+c = 33+32+26.48 = 91.48 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 91.48 }{ 2 } = 45.74 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 45.74 * (45.74-33)(45.74-32)(45.74-26.48) } ; ; T = sqrt{ 154186.07 } = 392.67 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 392.67 }{ 33 } = 23.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 392.67 }{ 32 } = 24.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 392.67 }{ 26.48 } = 29.66 ; ;

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{ 33**2-32**2-26.48**2 }{ 2 * 32 * 26.48 } ) = 67° 57'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32**2-33**2-26.48**2 }{ 2 * 33 * 26.48 } ) = 64° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 26.48**2-33**2-32**2 }{ 2 * 32 * 33 } ) = 48° 2'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 392.67 }{ 45.74 } = 8.58 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 33 }{ 2 * sin 67° 57'14" } = 17.8 ; ;





#2 Obtuse scalene triangle.

Sides: a = 33   b = 32   c = 2.45549062585

Area: T = 36.40765096341
Perimeter: p = 67.45549062585
Semiperimeter: s = 33.72774531292

Angle ∠ A = α = 112.0466220128° = 112°2'46″ = 1.95655754556 rad
Angle ∠ B = β = 64° = 1.11770107213 rad
Angle ∠ C = γ = 3.95437798724° = 3°57'14″ = 0.06990064767 rad

Height: ha = 2.20664551293
Height: hb = 2.27554068521
Height: hc = 29.66602035279

Median: ma = 15.58108626966
Median: mb = 17.07437600536
Median: mc = 32.48106613051

Inradius: r = 1.07994325173
Circumradius: R = 17.80216310476

Vertex coordinates: A[2.45549062585; 0] B[0; 0] C[14.4666247844; 29.66602035279]
Centroid: CG[5.64403847008; 9.88767345093]
Coordinates of the circumscribed circle: U[1.22774531292; 17.75992631258]
Coordinates of the inscribed circle: I[1.72774531292; 1.07994325173]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 67.95437798724° = 67°57'14″ = 1.95655754556 rad
∠ B' = β' = 116° = 1.11770107213 rad
∠ C' = γ' = 176.0466220128° = 176°2'46″ = 0.06990064767 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 33 ; ; b = 32 ; ; beta = 64° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 32**2 = 33**2 + c**2 -2 * 32 * c * cos (64° ) ; ; ; ; c**2 -28.932c +65 =0 ; ; p=1; q=-28.9324956881; r=65 ; ; D = q**2 - 4pr = 28.932**2 - 4 * 1 * 65 = 577.089306741 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 28.93 ± sqrt{ 577.09 } }{ 2 } ; ; c_{1,2} = 14.466247844 ± 12.0113415856 ; ; c_{1} = 26.4775894296 ; ;
c_{2} = 2.45490625847 ; ; ; ; (c -26.4775894296) (c -2.45490625847) = 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 = 33 ; ; b = 32 ; ; c = 2.45 ; ;

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

p = a+b+c = 33+32+2.45 = 67.45 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67.45 }{ 2 } = 33.73 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.73 * (33.73-33)(33.73-32)(33.73-2.45) } ; ; T = sqrt{ 1325.43 } = 36.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 36.41 }{ 33 } = 2.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 36.41 }{ 32 } = 2.28 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 36.41 }{ 2.45 } = 29.66 ; ;

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{ 33**2-32**2-2.45**2 }{ 2 * 32 * 2.45 } ) = 112° 2'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32**2-33**2-2.45**2 }{ 2 * 33 * 2.45 } ) = 64° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.45**2-33**2-32**2 }{ 2 * 32 * 33 } ) = 3° 57'14" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 36.41 }{ 33.73 } = 1.08 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 33 }{ 2 * sin 112° 2'46" } = 17.8 ; ;




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