Triangle calculator SSA

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Triangle has two solutions with side c=45.23333411107 and with side c=11.40875146193

#1 Acute scalene triangle.

Sides: a = 46   b = 40   c = 45.23333411107

Area: T = 819.8220261985
Perimeter: p = 131.2333341111
Semiperimeter: s = 65.61766705553

Angle ∠ A = α = 64.98770666671° = 64°59'13″ = 1.13442382846 rad
Angle ∠ B = β = 52° = 0.9087571211 rad
Angle ∠ C = γ = 63.01329333329° = 63°47″ = 1.1099783158 rad

Height: ha = 35.64443592168
Height: hb = 40.99110130993
Height: hc = 36.24884946659

Median: ma = 35.97325947635
Median: mb = 411.0003362671
Median: mc = 36.69444984022

Inradius: r = 12.49440850404
Circumradius: R = 25.38803643015

Vertex coordinates: A[45.23333411107; 0] B[0; 0] C[28.3220427865; 36.24884946659]
Centroid: CG[24.51879229919; 12.08328315553]
Coordinates of the circumscribed circle: U[22.61766705553; 11.51773393224]
Coordinates of the inscribed circle: I[25.61766705553; 12.49440850404]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115.0132933333° = 115°47″ = 1.13442382846 rad
∠ B' = β' = 128° = 0.9087571211 rad
∠ C' = γ' = 116.9877066667° = 116°59'13″ = 1.1099783158 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 = 46 ; ; b = 40 ; ; c = 45.23 ; ;

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

p = a+b+c = 46+40+45.23 = 131.23 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 131.23 }{ 2 } = 65.62 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 65.62 * (65.62-46)(65.62-40)(65.62-45.23) } ; ; T = sqrt{ 672105.26 } = 819.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 819.82 }{ 46 } = 35.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 819.82 }{ 40 } = 40.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 819.82 }{ 45.23 } = 36.25 ; ;

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{ 46**2-40**2-45.23**2 }{ 2 * 40 * 45.23 } ) = 64° 59'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 40**2-46**2-45.23**2 }{ 2 * 46 * 45.23 } ) = 52° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 45.23**2-46**2-40**2 }{ 2 * 40 * 46 } ) = 63° 47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 819.82 }{ 65.62 } = 12.49 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 46 }{ 2 * sin 64° 59'13" } = 25.38 ; ;





#2 Obtuse scalene triangle.

Sides: a = 46   b = 40   c = 11.40875146193

Area: T = 206.7532616415
Perimeter: p = 97.40875146193
Semiperimeter: s = 48.70437573097

Angle ∠ A = α = 115.0132933333° = 115°47″ = 2.0077354369 rad
Angle ∠ B = β = 52° = 0.9087571211 rad
Angle ∠ C = γ = 12.98770666671° = 12°59'13″ = 0.22766670735 rad

Height: ha = 8.98992441919
Height: hb = 10.33876308207
Height: hc = 36.24884946659

Median: ma = 18.33220946674
Median: mb = 26.89898809015
Median: mc = 42.72554859838

Inradius: r = 4.24551060829
Circumradius: R = 25.38803643015

Vertex coordinates: A[11.40875146193; 0] B[0; 0] C[28.3220427865; 36.24884946659]
Centroid: CG[13.24326474948; 12.08328315553]
Coordinates of the circumscribed circle: U[5.70437573097; 24.73111553436]
Coordinates of the inscribed circle: I[8.70437573097; 4.24551060829]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 64.98770666671° = 64°59'13″ = 2.0077354369 rad
∠ B' = β' = 128° = 0.9087571211 rad
∠ C' = γ' = 167.0132933333° = 167°47″ = 0.22766670735 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 46 ; ; b = 40 ; ; beta = 52° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 40**2 = 46**2 + c**2 -2 * 40 * c * cos (52° ) ; ; ; ; c**2 -56.641c +516 =0 ; ; p=1; q=-56.64085573; r=516 ; ; D = q**2 - 4pr = 56.641**2 - 4 * 1 * 516 = 1144.18653782 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 56.64 ± sqrt{ 1144.19 } }{ 2 } ; ; c_{1,2} = 28.320427865 ± 16.9129132457 ; ; c_{1} = 45.2333411107 ; ;
c_{2} = 11.4075146193 ; ; ; ; (c -45.2333411107) (c -11.4075146193) = 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 = 46 ; ; b = 40 ; ; c = 11.41 ; ;

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

p = a+b+c = 46+40+11.41 = 97.41 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 97.41 }{ 2 } = 48.7 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 48.7 * (48.7-46)(48.7-40)(48.7-11.41) } ; ; T = sqrt{ 42746.64 } = 206.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 206.75 }{ 46 } = 8.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 206.75 }{ 40 } = 10.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 206.75 }{ 11.41 } = 36.25 ; ;

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{ 46**2-40**2-11.41**2 }{ 2 * 40 * 11.41 } ) = 115° 47" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 40**2-46**2-11.41**2 }{ 2 * 46 * 11.41 } ) = 52° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.41**2-46**2-40**2 }{ 2 * 40 * 46 } ) = 12° 59'13" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 206.75 }{ 48.7 } = 4.25 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 46 }{ 2 * sin 115° 47" } = 25.38 ; ;




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