Triangle calculator SSA

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Triangle has two solutions with side c=293.1122296564 and with side c=13.30554806834

#1 Obtuse scalene triangle.

Sides: a = 200   b = 190   c = 293.1122296564

Area: T = 18840.89552478
Perimeter: p = 683.1122296564
Semiperimeter: s = 341.5566148282

Angle ∠ A = α = 42.58799653282° = 42°34'48″ = 0.74331605904 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 97.42200346718° = 97°25'12″ = 1.77003003624 rad

Height: ha = 188.4098952478
Height: hb = 198.3255213135
Height: hc = 128.5587521937

Median: ma = 225.8488199458
Median: mb = 232.2343522986
Median: mc = 128.732954362

Inradius: r = 55.16219267947
Circumradius: R = 147.7943763552

Vertex coordinates: A[293.1122296564; 0] B[0; 0] C[153.2098888624; 128.5587521937]
Centroid: CG[148.7743728396; 42.85325073124]
Coordinates of the circumscribed circle: U[146.5566148282; -19.08664335459]
Coordinates of the inscribed circle: I[151.5566148282; 55.16219267947]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.4220034672° = 137°25'12″ = 0.74331605904 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 82.58799653282° = 82°34'48″ = 1.77003003624 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 = 200 ; ; b = 190 ; ; c = 293.11 ; ;

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

p = a+b+c = 200+190+293.11 = 683.11 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 683.11 }{ 2 } = 341.56 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 341.56 * (341.56-200)(341.56-190)(341.56-293.11) } ; ; T = sqrt{ 354979333.74 } = 18840.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18840.9 }{ 200 } = 188.41 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18840.9 }{ 190 } = 198.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18840.9 }{ 293.11 } = 128.56 ; ;

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{ 200**2-190**2-293.11**2 }{ 2 * 190 * 293.11 } ) = 42° 34'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 190**2-200**2-293.11**2 }{ 2 * 200 * 293.11 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 293.11**2-200**2-190**2 }{ 2 * 190 * 200 } ) = 97° 25'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18840.9 }{ 341.56 } = 55.16 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 200 }{ 2 * sin 42° 34'48" } = 147.79 ; ;





#2 Obtuse scalene triangle.

Sides: a = 200   b = 190   c = 13.30554806834

Area: T = 855.2659812421
Perimeter: p = 403.3055480683
Semiperimeter: s = 201.6532740342

Angle ∠ A = α = 137.4220034672° = 137°25'12″ = 2.39884320632 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 2.58799653282° = 2°34'48″ = 0.04550288896 rad

Height: ha = 8.55325981242
Height: hb = 9.00327348676
Height: hc = 128.5587521937

Median: ma = 90.21437345869
Median: mb = 105.1833258687
Median: mc = 194.9510611812

Inradius: r = 4.2411250632
Circumradius: R = 147.7943763552

Vertex coordinates: A[13.30554806834; 0] B[0; 0] C[153.2098888624; 128.5587521937]
Centroid: CG[55.50547897691; 42.85325073124]
Coordinates of the circumscribed circle: U[6.65327403417; 147.6443955483]
Coordinates of the inscribed circle: I[11.65327403417; 4.2411250632]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 42.58799653282° = 42°34'48″ = 2.39884320632 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 177.4220034672° = 177°25'12″ = 0.04550288896 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 200 ; ; b = 190 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 190**2 = 200**2 + c**2 -2 * 190 * c * cos (40° ) ; ; ; ; c**2 -306.418c +3900 =0 ; ; p=1; q=-306.417777248; r=3900 ; ; D = q**2 - 4pr = 306.418**2 - 4 * 1 * 3900 = 78291.8542134 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 306.42 ± sqrt{ 78291.85 } }{ 2 } ; ; c_{1,2} = 153.208888624 ± 139.90340794 ; ;
c_{1} = 293.112296564 ; ; c_{2} = 13.3054806834 ; ; ; ; (c -293.112296564) (c -13.3054806834) = 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 = 200 ; ; b = 190 ; ; c = 13.31 ; ;

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

p = a+b+c = 200+190+13.31 = 403.31 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 403.31 }{ 2 } = 201.65 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 201.65 * (201.65-200)(201.65-190)(201.65-13.31) } ; ; T = sqrt{ 731469.35 } = 855.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 855.26 }{ 200 } = 8.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 855.26 }{ 190 } = 9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 855.26 }{ 13.31 } = 128.56 ; ;

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{ 200**2-190**2-13.31**2 }{ 2 * 190 * 13.31 } ) = 137° 25'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 190**2-200**2-13.31**2 }{ 2 * 200 * 13.31 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13.31**2-200**2-190**2 }{ 2 * 190 * 200 } ) = 2° 34'48" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 855.26 }{ 201.65 } = 4.24 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 200 }{ 2 * sin 137° 25'12" } = 147.79 ; ;




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