Triangle calculator SSA

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Triangle has two solutions with side c=103.2921895057 and with side c=11.5898518144

#1 Obtuse scalene triangle.

Sides: a = 71   b = 62   c = 103.2921895057

Area: T = 2155.328756716
Perimeter: p = 236.2921895057
Semiperimeter: s = 118.1465947529

Angle ∠ A = α = 42.30774684462° = 42°18'27″ = 0.73884046226 rad
Angle ∠ B = β = 36° = 0.62883185307 rad
Angle ∠ C = γ = 101.6932531554° = 101°41'33″ = 1.77548695003 rad

Height: ha = 60.7133452596
Height: hb = 69.52766957148
Height: hc = 41.73327529128

Median: ma = 77.43661530053
Median: mb = 83.03107641315
Median: mc = 42.13330761264

Inradius: r = 18.24329242157
Circumradius: R = 52.74403501178

Vertex coordinates: A[103.2921895057; 0] B[0; 0] C[57.44402066006; 41.73327529128]
Centroid: CG[53.57773672193; 13.91109176376]
Coordinates of the circumscribed circle: U[51.64659475286; -10.6888341051]
Coordinates of the inscribed circle: I[56.14659475286; 18.24329242157]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.6932531554° = 137°41'33″ = 0.73884046226 rad
∠ B' = β' = 144° = 0.62883185307 rad
∠ C' = γ' = 78.30774684462° = 78°18'27″ = 1.77548695003 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 71 ; ; b = 62 ; ; beta = 36° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 62**2 = 71**2 + c**2 -2 * 71 * c * cos (36° ) ; ; ; ; c**2 -114.88c +1197 =0 ; ; p=1; q=-114.88; r=1197 ; ; D = q**2 - 4pr = 114.88**2 - 4 * 1 * 1197 = 8409.50933729 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 114.88 ± sqrt{ 8409.51 } }{ 2 } ; ; c_{1,2} = 57.4402066 ± 45.8516884566 ; ; c_{1} = 103.291895057 ; ;
c_{2} = 11.5885181434 ; ; ; ; text{ Factored form: } ; ; (c -103.291895057) (c -11.5885181434) = 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 = 71 ; ; b = 62 ; ; c = 103.29 ; ;

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

p = a+b+c = 71+62+103.29 = 236.29 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 236.29 }{ 2 } = 118.15 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 118.15 * (118.15-71)(118.15-62)(118.15-103.29) } ; ; T = sqrt{ 4645436.92 } = 2155.33 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2155.33 }{ 71 } = 60.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2155.33 }{ 62 } = 69.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2155.33 }{ 103.29 } = 41.73 ; ;

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{ 62**2+103.29**2-71**2 }{ 2 * 62 * 103.29 } ) = 42° 18'27" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 71**2+103.29**2-62**2 }{ 2 * 71 * 103.29 } ) = 36° ; ; gamma = 180° - alpha - beta = 180° - 42° 18'27" - 36° = 101° 41'33" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2155.33 }{ 118.15 } = 18.24 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 71 }{ 2 * sin 42° 18'27" } = 52.74 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 103.29**2 - 71**2 } }{ 2 } = 77.436 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 103.29**2+2 * 71**2 - 62**2 } }{ 2 } = 83.031 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 71**2 - 103.29**2 } }{ 2 } = 42.133 ; ;







#2 Obtuse scalene triangle.

Sides: a = 71   b = 62   c = 11.5898518144

Area: T = 241.8110382165
Perimeter: p = 144.5898518144
Semiperimeter: s = 72.2944259072

Angle ∠ A = α = 137.6932531554° = 137°41'33″ = 2.4033188031 rad
Angle ∠ B = β = 36° = 0.62883185307 rad
Angle ∠ C = γ = 6.30774684462° = 6°18'27″ = 0.11100860919 rad

Height: ha = 6.8121560061
Height: hb = 7.88003349085
Height: hc = 41.73327529128

Median: ma = 26.99880902359
Median: mb = 40.33217105562
Median: mc = 66.43997482059

Inradius: r = 3.34548075306
Circumradius: R = 52.74403501178

Vertex coordinates: A[11.5898518144; 0] B[0; 0] C[57.44402066006; 41.73327529128]
Centroid: CG[23.01095749149; 13.91109176376]
Coordinates of the circumscribed circle: U[5.7944259072; 52.42110939638]
Coordinates of the inscribed circle: I[10.2944259072; 3.34548075306]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 42.30774684462° = 42°18'27″ = 2.4033188031 rad
∠ B' = β' = 144° = 0.62883185307 rad
∠ C' = γ' = 173.6932531554° = 173°41'33″ = 0.11100860919 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 71 ; ; b = 62 ; ; beta = 36° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 62**2 = 71**2 + c**2 -2 * 71 * c * cos (36° ) ; ; ; ; c**2 -114.88c +1197 =0 ; ; p=1; q=-114.88; r=1197 ; ; D = q**2 - 4pr = 114.88**2 - 4 * 1 * 1197 = 8409.50933729 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 114.88 ± sqrt{ 8409.51 } }{ 2 } ; ; c_{1,2} = 57.4402066 ± 45.8516884566 ; ; c_{1} = 103.291895057 ; ; : Nr. 1
c_{2} = 11.5885181434 ; ; ; ; text{ Factored form: } ; ; (c -103.291895057) (c -11.5885181434) = 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 = 71 ; ; b = 62 ; ; c = 11.59 ; ;

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

p = a+b+c = 71+62+11.59 = 144.59 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 144.59 }{ 2 } = 72.29 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 72.29 * (72.29-71)(72.29-62)(72.29-11.59) } ; ; T = sqrt{ 58472.26 } = 241.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 241.81 }{ 71 } = 6.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 241.81 }{ 62 } = 7.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 241.81 }{ 11.59 } = 41.73 ; ;

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{ 62**2+11.59**2-71**2 }{ 2 * 62 * 11.59 } ) = 137° 41'33" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 71**2+11.59**2-62**2 }{ 2 * 71 * 11.59 } ) = 36° ; ; gamma = 180° - alpha - beta = 180° - 137° 41'33" - 36° = 6° 18'27" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 241.81 }{ 72.29 } = 3.34 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 71 }{ 2 * sin 137° 41'33" } = 52.74 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 11.59**2 - 71**2 } }{ 2 } = 26.998 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.59**2+2 * 71**2 - 62**2 } }{ 2 } = 40.332 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 71**2 - 11.59**2 } }{ 2 } = 66.4 ; ;
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