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?

1. Use Law of Cosines

a = 350 ; ; b = 286 ; ; beta = 46° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 286**2 = 350**2 + c**2 -2 * 350 * c * cos (46° ) ; ; ; ; c**2 -486.261c +40704 =0 ; ; p=1; q=-486.261; 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} = 243.13042966 ± 135.677580414 ; ; c_{1} = 378.808010074 ; ; c_{2} = 107.452849246 ; ; ; ; text{ Factored form: } ; ; (c -378.808010074) (c -107.452849246) = 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 = 378.81 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 286**2+378.81**2-350**2 }{ 2 * 286 * 378.81 } ) = 61° 40'48" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 350**2+378.81**2-286**2 }{ 2 * 350 * 378.81 } ) = 46° ; ; gamma = 180° - alpha - beta = 180° - 61° 40'48" - 46° = 72° 19'12" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 286**2+2 * 378.81**2 - 350**2 } }{ 2 } = 286.393 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 378.81**2+2 * 350**2 - 286**2 } }{ 2 } = 335.483 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 286**2+2 * 350**2 - 378.81**2 } }{ 2 } = 257.438 ; ;







#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 - 2ac cos beta ; ; 286**2 = 350**2 + c**2 -2 * 350 * c * cos (46° ) ; ; ; ; c**2 -486.261c +40704 =0 ; ; p=1; q=-486.261; 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} = 243.13042966 ± 135.677580414 ; ; c_{1} = 378.808010074 ; ; c_{2} = 107.452849246 ; ; ; ; text{ Factored form: } ; ; (c -378.808010074) (c -107.452849246) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 286**2+107.45**2-350**2 }{ 2 * 286 * 107.45 } ) = 118° 19'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 350**2+107.45**2-286**2 }{ 2 * 350 * 107.45 } ) = 46° ; ; gamma = 180° - alpha - beta = 180° - 118° 19'12" - 46° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 286**2+2 * 107.45**2 - 350**2 } }{ 2 } = 126.673 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 107.45**2+2 * 350**2 - 286**2 } }{ 2 } = 215.81 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 286**2+2 * 350**2 - 107.45**2 } }{ 2 } = 315.058 ; ;
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