Triangle calculator SSA

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Triangle has two solutions with side c=378.8088010075 and with side c=107.4532849247

#1 Acute scalene triangle.

Sides: a = 350   b = 286   c = 378.8088010075

Area: T = 47686.04437084
Perimeter: p = 1014.808801007
Semiperimeter: s = 507.4044005037

Angle ∠ A = α = 61.68798952392° = 61°40'48″ = 1.07765172542 rad
Angle ∠ B = β = 46° = 0.80328514559 rad
Angle ∠ C = γ = 72.32201047608° = 72°19'12″ = 1.26222239435 rad

Height: ha = 272.4921678334
Height: hb = 333.4698837122
Height: hc = 251.7698930119

Median: ma = 286.3932657462
Median: mb = 335.4832867295
Median: mc = 257.4387609676

Inradius: r = 93.98804243463
Circumradius: R = 198.7933393515

Vertex coordinates: A[378.8088010075; 0] B[0; 0] C[243.1330429661; 251.7698930119]
Centroid: CG[207.3132813245; 83.92329767062]
Coordinates of the circumscribed circle: U[189.4044005037; 60.37333068599]
Coordinates of the inscribed circle: I[221.4044005037; 93.98804243463]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 118.3220104761° = 118°19'12″ = 1.07765172542 rad
∠ B' = β' = 134° = 0.80328514559 rad
∠ C' = γ' = 107.6879895239° = 107°40'48″ = 1.26222239435 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 = 350 ; ; b = 286 ; ; c = 378.81 ; ;

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

p = a+b+c = 350+286+378.81 = 1014.81 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1014.81 }{ 2 } = 507.4 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 507.4 * (507.4-350)(507.4-286)(507.4-378.81) } ; ; T = sqrt{ 2273958764.56 } = 47686.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 47686.04 }{ 350 } = 272.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 47686.04 }{ 286 } = 333.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 47686.04 }{ 378.81 } = 251.77 ; ;

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{ 350**2-286**2-378.81**2 }{ 2 * 286 * 378.81 } ) = 61° 40'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 286**2-350**2-378.81**2 }{ 2 * 350 * 378.81 } ) = 46° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 378.81**2-350**2-286**2 }{ 2 * 286 * 350 } ) = 72° 19'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 47686.04 }{ 507.4 } = 93.98 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 350 }{ 2 * sin 61° 40'48" } = 198.79 ; ;





#2 Obtuse scalene triangle.

Sides: a = 350   b = 286   c = 107.4532849247

Area: T = 13526.64444465
Perimeter: p = 743.4532849247
Semiperimeter: s = 371.7266424623

Angle ∠ A = α = 118.3220104761° = 118°19'12″ = 2.06550753994 rad
Angle ∠ B = β = 46° = 0.80328514559 rad
Angle ∠ C = γ = 15.68798952392° = 15°40'48″ = 0.27436657983 rad

Height: ha = 77.2955111123
Height: hb = 94.59219192064
Height: hc = 251.7698930119

Median: ma = 126.6733033459
Median: mb = 215.811023471
Median: mc = 315.0587885629

Inradius: r = 36.38987083363
Circumradius: R = 198.7933393515

Vertex coordinates: A[107.4532849247; 0] B[0; 0] C[243.1330429661; 251.7698930119]
Centroid: CG[116.8611092969; 83.92329767062]
Coordinates of the circumscribed circle: U[53.72664246234; 191.3965623259]
Coordinates of the inscribed circle: I[85.72664246234; 36.38987083363]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 61.68798952392° = 61°40'48″ = 2.06550753994 rad
∠ B' = β' = 134° = 0.80328514559 rad
∠ C' = γ' = 164.3220104761° = 164°19'12″ = 0.27436657983 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 350 ; ; b = 286 ; ; beta = 46° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 286**2 = 350**2 + c**2 -2 * 286 * c * cos (46° ) ; ; ; ; c**2 -486.261c +40704 =0 ; ; p=1; q=-486.260859321; r=40704 ; ; D = q**2 - 4pr = 486.261**2 - 4 * 1 * 40704 = 73633.6233079 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 486.26 ± sqrt{ 73633.62 } }{ 2 } ; ; c_{1,2} = 243.130429661 ± 135.677580414 ; ;
c_{1} = 378.808010075 ; ; c_{2} = 107.452849247 ; ; ; ; (c -378.808010075) (c -107.452849247) = 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 = 350 ; ; b = 286 ; ; c = 107.45 ; ;

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

p = a+b+c = 350+286+107.45 = 743.45 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 743.45 }{ 2 } = 371.73 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 371.73 * (371.73-350)(371.73-286)(371.73-107.45) } ; ; T = sqrt{ 182970109.98 } = 13526.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13526.64 }{ 350 } = 77.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13526.64 }{ 286 } = 94.59 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13526.64 }{ 107.45 } = 251.77 ; ;

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{ 350**2-286**2-107.45**2 }{ 2 * 286 * 107.45 } ) = 118° 19'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 286**2-350**2-107.45**2 }{ 2 * 350 * 107.45 } ) = 46° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 107.45**2-350**2-286**2 }{ 2 * 286 * 350 } ) = 15° 40'48" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13526.64 }{ 371.73 } = 36.39 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 350 }{ 2 * sin 118° 19'12" } = 198.79 ; ;




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