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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 270 ; ; b = 260 ; ; beta = 73° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 260**2 = 270**2 + c**2 -2 * 260 * c * cos (73° ) ; ; ; ; c**2 -157.881c +5300 =0 ; ; p=1; q=-157.88072055; 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} = 78.9403602751 ± 30.5218033604 ; ; c_{1} = 109.462163636 ; ;
c_{2} = 48.4185569148 ; ; ; ; (c -109.462163636) (c -48.4185569148) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 270**2-260**2-48.42**2 }{ 2 * 260 * 48.42 } ) = 96° 44'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 260**2-270**2-48.42**2 }{ 2 * 270 * 48.42 } ) = 73° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 48.42**2-270**2-260**2 }{ 2 * 260 * 270 } ) = 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 ; ;




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