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?

1. Use Law of Cosines

a = 200 ; ; b = 190 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 190**2 = 200**2 + c**2 -2 * 200 * c * cos (40° ) ; ; ; ; c**2 -306.418c +3900 =0 ; ; p=1; q=-306.418; 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} = 153.20888862 ± 139.90340794 ; ; c_{1} = 293.11229656 ; ; c_{2} = 13.3054806796 ; ; ; ; text{ Factored form: } ; ; (c -293.11229656) (c -13.3054806796) = 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 = 293.11 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 190**2+293.11**2-200**2 }{ 2 * 190 * 293.11 } ) = 42° 34'48" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 200**2+293.11**2-190**2 }{ 2 * 200 * 293.11 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 42° 34'48" - 40° = 97° 25'12" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 190**2+2 * 293.11**2 - 200**2 } }{ 2 } = 225.848 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 293.11**2+2 * 200**2 - 190**2 } }{ 2 } = 232.234 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 190**2+2 * 200**2 - 293.11**2 } }{ 2 } = 128.73 ; ;







#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 200 ; ; b = 190 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 190**2 = 200**2 + c**2 -2 * 200 * c * cos (40° ) ; ; ; ; c**2 -306.418c +3900 =0 ; ; p=1; q=-306.418; 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} = 153.20888862 ± 139.90340794 ; ; c_{1} = 293.11229656 ; ; c_{2} = 13.3054806796 ; ; ; ; text{ Factored form: } ; ; (c -293.11229656) (c -13.3054806796) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 190**2+13.31**2-200**2 }{ 2 * 190 * 13.31 } ) = 137° 25'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 200**2+13.31**2-190**2 }{ 2 * 200 * 13.31 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 137° 25'12" - 40° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 190**2+2 * 13.31**2 - 200**2 } }{ 2 } = 90.214 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.31**2+2 * 200**2 - 190**2 } }{ 2 } = 105.183 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 190**2+2 * 200**2 - 13.31**2 } }{ 2 } = 194.951 ; ;
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