Triangle calculator SSA

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Triangle has two solutions with side c=94.51550067534 and with side c=46.55334537968

#1 Acute scalene triangle.

Sides: a = 120   b = 100   c = 94.51550067534

Area: T = 4587.855480122
Perimeter: p = 314.5155006753
Semiperimeter: s = 157.2587503377

Angle ∠ A = α = 76.12548052335° = 76°7'29″ = 1.32986284938 rad
Angle ∠ B = β = 54° = 0.94224777961 rad
Angle ∠ C = γ = 49.87551947665° = 49°52'31″ = 0.87704863637 rad

Height: ha = 76.46442466869
Height: hb = 91.75770960243
Height: hc = 97.0822039325

Median: ma = 76.59333629683
Median: mb = 95.74220662551
Median: mc = 99.83435032672

Inradius: r = 29.17441551449
Circumradius: R = 61.8033398875

Vertex coordinates: A[94.51550067534; 0] B[0; 0] C[70.53442302751; 97.0822039325]
Centroid: CG[55.01664123428; 32.3610679775]
Coordinates of the circumscribed circle: U[47.25875033767; 39.82994926795]
Coordinates of the inscribed circle: I[57.25875033767; 29.17441551449]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103.8755194767° = 103°52'31″ = 1.32986284938 rad
∠ B' = β' = 126° = 0.94224777961 rad
∠ C' = γ' = 130.1254805233° = 130°7'29″ = 0.87704863637 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 120 ; ; b = 100 ; ; beta = 54° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 100**2 = 120**2 + c**2 -2 * 120 * c * cos (54° ) ; ; ; ; c**2 -141.068c +4400 =0 ; ; p=1; q=-141.068; r=4400 ; ; D = q**2 - 4pr = 141.068**2 - 4 * 1 * 4400 = 2300.310562 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 141.07 ± sqrt{ 2300.31 } }{ 2 } ; ;
c_{1,2} = 70.53423028 ± 23.9807764783 ; ; c_{1} = 94.5150067534 ; ; c_{2} = 46.5534537968 ; ; ; ; text{ Factored form: } ; ; (c -94.5150067534) (c -46.5534537968) = 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 = 120 ; ; b = 100 ; ; c = 94.52 ; ;

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

p = a+b+c = 120+100+94.52 = 314.52 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 314.52 }{ 2 } = 157.26 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 157.26 * (157.26-120)(157.26-100)(157.26-94.52) } ; ; T = sqrt{ 21048411.68 } = 4587.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4587.85 }{ 120 } = 76.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4587.85 }{ 100 } = 91.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4587.85 }{ 94.52 } = 97.08 ; ;

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{ 100**2+94.52**2-120**2 }{ 2 * 100 * 94.52 } ) = 76° 7'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 120**2+94.52**2-100**2 }{ 2 * 120 * 94.52 } ) = 54° ; ;
 gamma = 180° - alpha - beta = 180° - 76° 7'29" - 54° = 49° 52'31" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4587.85 }{ 157.26 } = 29.17 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 120 }{ 2 * sin 76° 7'29" } = 61.8 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 94.52**2 - 120**2 } }{ 2 } = 76.593 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 94.52**2+2 * 120**2 - 100**2 } }{ 2 } = 95.742 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 120**2 - 94.52**2 } }{ 2 } = 99.834 ; ;



#2 Obtuse scalene triangle.

Sides: a = 120   b = 100   c = 46.55334537968

Area: T = 2259.752211611
Perimeter: p = 266.5533453797
Semiperimeter: s = 133.2776726898

Angle ∠ A = α = 103.8755194767° = 103°52'31″ = 1.81329641598 rad
Angle ∠ B = β = 54° = 0.94224777961 rad
Angle ∠ C = γ = 22.12548052335° = 22°7'29″ = 0.38661506977 rad

Height: ha = 37.66325352685
Height: hb = 45.19550423222
Height: hc = 97.0822039325

Median: ma = 49.83658508526
Median: mb = 76.05500626575
Median: mc = 107.9733116955

Inradius: r = 16.95553392306
Circumradius: R = 61.8033398875

Vertex coordinates: A[46.55334537968; 0] B[0; 0] C[70.53442302751; 97.0822039325]
Centroid: CG[39.0299228024; 32.3610679775]
Coordinates of the circumscribed circle: U[23.27767268984; 57.25325466455]
Coordinates of the inscribed circle: I[33.27767268984; 16.95553392306]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 76.12548052335° = 76°7'29″ = 1.81329641598 rad
∠ B' = β' = 126° = 0.94224777961 rad
∠ C' = γ' = 157.8755194767° = 157°52'31″ = 0.38661506977 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 120 ; ; b = 100 ; ; beta = 54° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 100**2 = 120**2 + c**2 -2 * 120 * c * cos (54° ) ; ; ; ; c**2 -141.068c +4400 =0 ; ; p=1; q=-141.068; r=4400 ; ; D = q**2 - 4pr = 141.068**2 - 4 * 1 * 4400 = 2300.310562 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 141.07 ± sqrt{ 2300.31 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 70.53423028 ± 23.9807764783 ; ; c_{1} = 94.5150067534 ; ; c_{2} = 46.5534537968 ; ; ; ; text{ Factored form: } ; ; (c -94.5150067534) (c -46.5534537968) = 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 = 120 ; ; b = 100 ; ; c = 46.55 ; ;

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

p = a+b+c = 120+100+46.55 = 266.55 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 266.55 }{ 2 } = 133.28 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 133.28 * (133.28-120)(133.28-100)(133.28-46.55) } ; ; T = sqrt{ 5106479.63 } = 2259.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2259.75 }{ 120 } = 37.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2259.75 }{ 100 } = 45.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2259.75 }{ 46.55 } = 97.08 ; ;

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{ 100**2+46.55**2-120**2 }{ 2 * 100 * 46.55 } ) = 103° 52'31" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 120**2+46.55**2-100**2 }{ 2 * 120 * 46.55 } ) = 54° ; ;
 gamma = 180° - alpha - beta = 180° - 103° 52'31" - 54° = 22° 7'29" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2259.75 }{ 133.28 } = 16.96 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 120 }{ 2 * sin 103° 52'31" } = 61.8 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 46.55**2 - 120**2 } }{ 2 } = 49.836 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 46.55**2+2 * 120**2 - 100**2 } }{ 2 } = 76.05 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 120**2 - 46.55**2 } }{ 2 } = 107.973 ; ;
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