Triangle calculator SSA

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Triangle has two solutions with side c=109.4622163636 and with side c=48.41985569148

#1 Acute scalene triangle.

Sides: a = 270   b = 260   c = 109.4622163636

Area: T = 14131.69903372
Perimeter: p = 639.4622163636
Semiperimeter: s = 319.7311081818

Angle ∠ A = α = 83.25884146181° = 83°15'30″ = 1.45331334651 rad
Angle ∠ B = β = 73° = 1.2744090354 rad
Angle ∠ C = γ = 23.74215853819° = 23°44'30″ = 0.41443688346 rad

Height: ha = 104.6799187683
Height: hb = 108.7055310286
Height: hc = 258.202228411

Median: ma = 146.8543609536
Median: mb = 159.8155464314
Median: mc = 259.335474253

Inradius: r = 44.19986755145
Circumradius: R = 135.9439928343

Vertex coordinates: A[109.4622163636; 0] B[0; 0] C[78.94403602751; 258.202228411]
Centroid: CG[62.80108413035; 86.06774280367]
Coordinates of the circumscribed circle: U[54.73110818178; 124.4355416185]
Coordinates of the inscribed circle: I[59.73110818178; 44.19986755145]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 96.74215853819° = 96°44'30″ = 1.45331334651 rad
∠ B' = β' = 107° = 1.2744090354 rad
∠ C' = γ' = 156.2588414618° = 156°15'30″ = 0.41443688346 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 270 ; ; b = 260 ; ; beta = 73° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 260**2 = 270**2 + c**2 -2 * 270 * c * cos (73° ) ; ; ; ; c**2 -157.881c +5300 =0 ; ; p=1; q=-157.881; r=5300 ; ; D = q**2 - 4pr = 157.881**2 - 4 * 1 * 5300 = 3726.32192147 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 157.88 ± sqrt{ 3726.32 } }{ 2 } ; ; c_{1,2} = 78.94036028 ± 30.5218033604 ; ; c_{1} = 109.46216364 ; ; c_{2} = 48.4185569196 ; ; ; ; text{ Factored form: } ; ; (c -109.46216364) (c -48.4185569196) = 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 = 270 ; ; b = 260 ; ; c = 109.46 ; ;

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

p = a+b+c = 270+260+109.46 = 639.46 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 639.46 }{ 2 } = 319.73 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 319.73 * (319.73-270)(319.73-260)(319.73-109.46) } ; ; T = sqrt{ 199704671.79 } = 14131.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14131.69 }{ 270 } = 104.68 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14131.69 }{ 260 } = 108.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14131.69 }{ 109.46 } = 258.2 ; ;

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{ 260**2+109.46**2-270**2 }{ 2 * 260 * 109.46 } ) = 83° 15'30" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 270**2+109.46**2-260**2 }{ 2 * 270 * 109.46 } ) = 73° ; ; gamma = 180° - alpha - beta = 180° - 83° 15'30" - 73° = 23° 44'30" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14131.69 }{ 319.73 } = 44.2 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 270 }{ 2 * sin 83° 15'30" } = 135.94 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 260**2+2 * 109.46**2 - 270**2 } }{ 2 } = 146.854 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 109.46**2+2 * 270**2 - 260**2 } }{ 2 } = 159.815 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 260**2+2 * 270**2 - 109.46**2 } }{ 2 } = 259.335 ; ;







#2 Obtuse scalene triangle.

Sides: a = 270   b = 260   c = 48.41985569148

Area: T = 6250.891099435
Perimeter: p = 578.4198556915
Semiperimeter: s = 289.2099278457

Angle ∠ A = α = 96.74215853819° = 96°44'30″ = 1.68884591885 rad
Angle ∠ B = β = 73° = 1.2744090354 rad
Angle ∠ C = γ = 10.25884146181° = 10°15'30″ = 0.17990431111 rad

Height: ha = 46.30328962545
Height: hb = 48.08437768796
Height: hc = 258.202228411

Median: ma = 129.4110889522
Median: mb = 143.9522000079
Median: mc = 263.9399218072

Inradius: r = 21.61437290881
Circumradius: R = 135.9439928343

Vertex coordinates: A[48.41985569148; 0] B[0; 0] C[78.94403602751; 258.202228411]
Centroid: CG[42.45329723966; 86.06774280367]
Coordinates of the circumscribed circle: U[24.20992784574; 133.7676867925]
Coordinates of the inscribed circle: I[29.20992784574; 21.61437290881]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 83.25884146181° = 83°15'30″ = 1.68884591885 rad
∠ B' = β' = 107° = 1.2744090354 rad
∠ C' = γ' = 169.7421585382° = 169°44'30″ = 0.17990431111 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 270 ; ; b = 260 ; ; beta = 73° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 260**2 = 270**2 + c**2 -2 * 270 * c * cos (73° ) ; ; ; ; c**2 -157.881c +5300 =0 ; ; p=1; q=-157.881; r=5300 ; ; D = q**2 - 4pr = 157.881**2 - 4 * 1 * 5300 = 3726.32192147 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 157.88 ± sqrt{ 3726.32 } }{ 2 } ; ; c_{1,2} = 78.94036028 ± 30.5218033604 ; ; c_{1} = 109.46216364 ; ; c_{2} = 48.4185569196 ; ; ; ; text{ Factored form: } ; ; (c -109.46216364) (c -48.4185569196) = 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 = 270 ; ; b = 260 ; ; c = 48.42 ; ;

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

p = a+b+c = 270+260+48.42 = 578.42 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 578.42 }{ 2 } = 289.21 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 289.21 * (289.21-270)(289.21-260)(289.21-48.42) } ; ; T = sqrt{ 39073638.22 } = 6250.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6250.89 }{ 270 } = 46.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6250.89 }{ 260 } = 48.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6250.89 }{ 48.42 } = 258.2 ; ;

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{ 260**2+48.42**2-270**2 }{ 2 * 260 * 48.42 } ) = 96° 44'30" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 270**2+48.42**2-260**2 }{ 2 * 270 * 48.42 } ) = 73° ; ; gamma = 180° - alpha - beta = 180° - 96° 44'30" - 73° = 10° 15'30" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6250.89 }{ 289.21 } = 21.61 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 270 }{ 2 * sin 96° 44'30" } = 135.94 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 260**2+2 * 48.42**2 - 270**2 } }{ 2 } = 129.411 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 48.42**2+2 * 270**2 - 260**2 } }{ 2 } = 143.952 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 260**2+2 * 270**2 - 48.42**2 } }{ 2 } = 263.939 ; ;
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