Triangle calculator SSA

Triangle has two solutions with side c=63.77107838668 and with side c=10.29326443773

#1 Obtuse scalene triangle.

Sides: a = 38.1 b = 28.2 c = 63.77107838668

Area: T = 285.6598500424
Perimeter: p = 130.0710783867
Semiperimeter: s = 65.03553919334

Angle ∠ A = α = 18.52334043434° = 18°31'24″ = 0.32332943945 rad
Angle ∠ B = β = 13.6° = 13°36' = 0.23773647783 rad
Angle ∠ C = γ = 147.8776595657° = 147°52'36″ = 2.58109334808 rad

Height: h_a = 14.99551968727
Height: h_b = 20.25994681152
Height: h_c = 8.95989145092

Median: m_a = 45.47660809382
Median: m_b = 50.65999153902
Median: m_c = 10.33218333926

Inradius: r = 4.39223545616
Circumradius: R = 59.96437377327

Vertex coordinates: A[63.77107838668; 0] B[0; 0] C[37.0321714122; 8.95989145092]
Centroid: CG[33.6010832663; 2.98663048364]
Coordinates of the circumscribed circle: U[31.88553919334; -50.78435763228]
Coordinates of the inscribed circle: I[36.83553919334; 4.39223545616]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.4776595657° = 161°28'36″ = 0.32332943945 rad
∠ B' = β' = 166.4° = 166°24' = 0.23773647783 rad
∠ C' = γ' = 32.12334043434° = 32°7'24″ = 2.58109334808 rad

How did we calculate this triangle?

1. Use Law of Cosines

$a = 38.1 ; ; b = 28.2 ; ; beta = 13° 36' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 28.2**2 = 38.1**2 + c**2 -2 * 38.1 * c * cos (13° 36') ; ; ; ; c**2 -74.063c +656.37 =0 ; ; p=1; q=-74.063; r=656.37 ; ; D = q**2 - 4pr = 74.063**2 - 4 * 1 * 656.37 = 2859.91140327 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 74.06 ± sqrt{ 2859.91 } }{ 2 } ; ; c_{1,2} = 37.03171412 ± 26.7390697448 ; ;$
$c_{1} = 63.7707838648 ; ; c_{2} = 10.2926443752 ; ; ; ; (c -63.7707838648) (c -10.2926443752) = 0 ; ; ; ; c>0 ; ;$

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.

$a = 38.1 ; ; b = 28.2 ; ; c = 63.77 ; ;$

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

$p = a+b+c = 38.1+28.2+63.77 = 130.07 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 130.07 }{ 2 } = 65.04 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 65.04 * (65.04-38.1)(65.04-28.2)(65.04-63.77) } ; ; T = sqrt{ 81600.78 } = 285.66 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 285.66 }{ 38.1 } = 15 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 285.66 }{ 28.2 } = 20.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 285.66 }{ 63.77 } = 8.96 ; ;$

6. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 38.1**2-28.2**2-63.77**2 }{ 2 * 28.2 * 63.77 } ) = 18° 31'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28.2**2-38.1**2-63.77**2 }{ 2 * 38.1 * 63.77 } ) = 13° 36' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 63.77**2-38.1**2-28.2**2 }{ 2 * 28.2 * 38.1 } ) = 147° 52'36" ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 285.66 }{ 65.04 } = 4.39 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 38.1 }{ 2 * sin 18° 31'24" } = 59.96 ; ;$

#2 Obtuse scalene triangle.

Sides: a = 38.1 b = 28.2 c = 10.29326443773

Area: T = 46.10554605248
Perimeter: p = 76.59326443773
Semiperimeter: s = 38.29663221886

Angle ∠ A = α = 161.4776595657° = 161°28'36″ = 2.81882982591 rad
Angle ∠ B = β = 13.6° = 13°36' = 0.23773647783 rad
Angle ∠ C = γ = 4.92334043434° = 4°55'24″ = 0.08659296162 rad

Height: h_a = 2.42202341483
Height: h_b = 3.27698908174
Height: h_c = 8.95989145092

Median: m_a = 9.36441211087
Median: m_b = 24.08224472207
Median: m_c = 33.12200900955

Inradius: r = 1.20439135324
Circumradius: R = 59.96437377327

Vertex coordinates: A[10.29326443773; 0] B[0; 0] C[37.0321714122; 8.95989145092]
Centroid: CG[15.77547861664; 2.98663048364]
Coordinates of the circumscribed circle: U[5.14663221886; 59.7422490832]
Coordinates of the inscribed circle: I[10.09663221886; 1.20439135324]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 18.52334043434° = 18°31'24″ = 2.81882982591 rad
∠ B' = β' = 166.4° = 166°24' = 0.23773647783 rad
∠ C' = γ' = 175.0776595657° = 175°4'36″ = 0.08659296162 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

$a = 38.1 ; ; b = 28.2 ; ; beta = 13° 36' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 28.2**2 = 38.1**2 + c**2 -2 * 38.1 * c * cos (13° 36') ; ; ; ; c**2 -74.063c +656.37 =0 ; ; p=1; q=-74.063; r=656.37 ; ; D = q**2 - 4pr = 74.063**2 - 4 * 1 * 656.37 = 2859.91140327 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 74.06 ± sqrt{ 2859.91 } }{ 2 } ; ; c_{1,2} = 37.03171412 ± 26.7390697448 ; ; : Nr. 1$
$c_{1} = 63.7707838648 ; ; c_{2} = 10.2926443752 ; ; ; ; (c -63.7707838648) (c -10.2926443752) = 0 ; ; ; ; c>0 ; ; : Nr. 1$

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.

$a = 38.1 ; ; b = 28.2 ; ; c = 10.29 ; ;$

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

$p = a+b+c = 38.1+28.2+10.29 = 76.59 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 76.59 }{ 2 } = 38.3 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.3 * (38.3-38.1)(38.3-28.2)(38.3-10.29) } ; ; T = sqrt{ 2125.71 } = 46.11 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 46.11 }{ 38.1 } = 2.42 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 46.11 }{ 28.2 } = 3.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 46.11 }{ 10.29 } = 8.96 ; ;$

6. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 38.1**2-28.2**2-10.29**2 }{ 2 * 28.2 * 10.29 } ) = 161° 28'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28.2**2-38.1**2-10.29**2 }{ 2 * 38.1 * 10.29 } ) = 13° 36' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.29**2-38.1**2-28.2**2 }{ 2 * 28.2 * 38.1 } ) = 4° 55'24" ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 46.11 }{ 38.3 } = 1.2 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 38.1 }{ 2 * sin 161° 28'36" } = 59.96 ; ;$

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