Triangle calculator SSA

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Triangle has two solutions with side c=14.56223604111 and with side c=0.81331992112

#1 Obtuse scalene triangle.

Sides: a = 7.89   b = 7.1   c = 14.56223604111

Area: T = 12.92331033048
Perimeter: p = 29.55223604111
Semiperimeter: s = 14.77661802055

Angle ∠ A = α = 14.47663771888° = 14°28'35″ = 0.25326604457 rad
Angle ∠ B = β = 13° = 0.22768928028 rad
Angle ∠ C = γ = 152.5243622811° = 152°31'25″ = 2.66220394051 rad

Height: ha = 3.27658183282
Height: hb = 3.64403107901
Height: hc = 1.77548638188

Median: ma = 10.75551450651
Median: mb = 11.16604086113
Median: mc = 1.82108417873

Inradius: r = 0.87545902612
Circumradius: R = 15.78112107632

Vertex coordinates: A[14.56223604111; 0] B[0; 0] C[7.68877798112; 1.77548638188]
Centroid: CG[7.41767134074; 0.59216212729]
Coordinates of the circumscribed circle: U[7.28111802055; -14.00111080978]
Coordinates of the inscribed circle: I[7.67661802055; 0.87545902612]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.5243622811° = 165°31'25″ = 0.25326604457 rad
∠ B' = β' = 167° = 0.22768928028 rad
∠ C' = γ' = 27.47663771888° = 27°28'35″ = 2.66220394051 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 7.89 ; ; b = 7.1 ; ; beta = 13° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 7.1**2 = 7.89**2 + c**2 -2 * 7.89 * c * cos (13° ) ; ; ; ; c**2 -15.376c +11.842 =0 ; ; p=1; q=-15.376; r=11.842 ; ; D = q**2 - 4pr = 15.376**2 - 4 * 1 * 11.842 = 189.039433699 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.38 ± sqrt{ 189.04 } }{ 2 } ; ; c_{1,2} = 7.68777981 ± 6.87458059992 ; ; c_{1} = 14.5623604099 ; ;
c_{2} = 0.813199210079 ; ; ; ; text{ Factored form: } ; ; (c -14.5623604099) (c -0.813199210079) = 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 = 7.89 ; ; b = 7.1 ; ; c = 14.56 ; ;

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

p = a+b+c = 7.89+7.1+14.56 = 29.55 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29.55 }{ 2 } = 14.78 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.78 * (14.78-7.89)(14.78-7.1)(14.78-14.56) } ; ; T = sqrt{ 167.01 } = 12.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12.92 }{ 7.89 } = 3.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12.92 }{ 7.1 } = 3.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12.92 }{ 14.56 } = 1.77 ; ;

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{ 7.1**2+14.56**2-7.89**2 }{ 2 * 7.1 * 14.56 } ) = 14° 28'35" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.89**2+14.56**2-7.1**2 }{ 2 * 7.89 * 14.56 } ) = 13° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 7.89**2+7.1**2-14.56**2 }{ 2 * 7.89 * 7.1 } ) = 152° 31'25" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12.92 }{ 14.78 } = 0.87 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.89 }{ 2 * sin 14° 28'35" } = 15.78 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.1**2+2 * 14.56**2 - 7.89**2 } }{ 2 } = 10.755 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.56**2+2 * 7.89**2 - 7.1**2 } }{ 2 } = 11.16 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.1**2+2 * 7.89**2 - 14.56**2 } }{ 2 } = 1.821 ; ;







#2 Obtuse scalene triangle.

Sides: a = 7.89   b = 7.1   c = 0.81331992112

Area: T = 0.72216589287
Perimeter: p = 15.80331992112
Semiperimeter: s = 7.90215996056

Angle ∠ A = α = 165.5243622811° = 165°31'25″ = 2.88989322079 rad
Angle ∠ B = β = 13° = 0.22768928028 rad
Angle ∠ C = γ = 1.47663771888° = 1°28'35″ = 0.02657676429 rad

Height: ha = 0.183293002
Height: hb = 0.20332842053
Height: hc = 1.77548638188

Median: ma = 3.15879457688
Median: mb = 4.34221419229
Median: mc = 7.49443796782

Inradius: r = 0.09113307387
Circumradius: R = 15.78112107632

Vertex coordinates: A[0.81331992112; 0] B[0; 0] C[7.68877798112; 1.77548638188]
Centroid: CG[2.83436596741; 0.59216212729]
Coordinates of the circumscribed circle: U[0.40765996056; 15.77659719166]
Coordinates of the inscribed circle: I[0.80215996056; 0.09113307387]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 14.47663771888° = 14°28'35″ = 2.88989322079 rad
∠ B' = β' = 167° = 0.22768928028 rad
∠ C' = γ' = 178.5243622811° = 178°31'25″ = 0.02657676429 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 7.89 ; ; b = 7.1 ; ; beta = 13° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 7.1**2 = 7.89**2 + c**2 -2 * 7.89 * c * cos (13° ) ; ; ; ; c**2 -15.376c +11.842 =0 ; ; p=1; q=-15.376; r=11.842 ; ; D = q**2 - 4pr = 15.376**2 - 4 * 1 * 11.842 = 189.039433699 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.38 ± sqrt{ 189.04 } }{ 2 } ; ; c_{1,2} = 7.68777981 ± 6.87458059992 ; ; c_{1} = 14.5623604099 ; ; : Nr. 1
c_{2} = 0.813199210079 ; ; ; ; text{ Factored form: } ; ; (c -14.5623604099) (c -0.813199210079) = 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 = 7.89 ; ; b = 7.1 ; ; c = 0.81 ; ;

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

p = a+b+c = 7.89+7.1+0.81 = 15.8 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.8 }{ 2 } = 7.9 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.9 * (7.9-7.89)(7.9-7.1)(7.9-0.81) } ; ; T = sqrt{ 0.52 } = 0.72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.72 }{ 7.89 } = 0.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.72 }{ 7.1 } = 0.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.72 }{ 0.81 } = 1.77 ; ;

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{ 7.1**2+0.81**2-7.89**2 }{ 2 * 7.1 * 0.81 } ) = 165° 31'25" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.89**2+0.81**2-7.1**2 }{ 2 * 7.89 * 0.81 } ) = 13° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 7.89**2+7.1**2-0.81**2 }{ 2 * 7.89 * 7.1 } ) = 1° 28'35" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.72 }{ 7.9 } = 0.09 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.89 }{ 2 * sin 165° 31'25" } = 15.78 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.1**2+2 * 0.81**2 - 7.89**2 } }{ 2 } = 3.158 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.81**2+2 * 7.89**2 - 7.1**2 } }{ 2 } = 4.342 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.1**2+2 * 7.89**2 - 0.81**2 } }{ 2 } = 7.494 ; ;
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