Triangle calculator SSA

Triangle has two solutions with side c=112.6298611148 and with side c=71.2166451293

#1 Acute scalene triangle.

Sides: a = 105.1 b = 55 c = 112.6298611148

Area: T = 2869.411046718
Perimeter: p = 272.7298611148
Semiperimeter: s = 136.3644305574

Angle ∠ A = α = 67.88545444666° = 67°53'4″ = 1.18548088122 rad
Angle ∠ B = β = 29° = 0.50661454831 rad
Angle ∠ C = γ = 83.11554555334° = 83°6'56″ = 1.45106383584 rad

Height: h_a = 54.60334341994
Height: h_b = 104.3422198806
Height: h_c = 50.95334910879

Median: m_a = 71.36994579251
Median: m_b = 105.4010934647
Median: m_c = 62.16327218495

Inradius: r = 21.04222401603
Circumradius: R = 56.72332968398

Vertex coordinates: A[112.6298611148; 0] B[0; 0] C[91.92325312204; 50.95334910879]
Centroid: CG[68.18437141227; 16.98444970293]
Coordinates of the circumscribed circle: U[56.31443055738; 6.79993670373]
Coordinates of the inscribed circle: I[81.36443055738; 21.04222401603]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.1155455533° = 112°6'56″ = 1.18548088122 rad
∠ B' = β' = 151° = 0.50661454831 rad
∠ C' = γ' = 96.88545444666° = 96°53'4″ = 1.45106383584 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.

$a = 105.1 ; ; b = 55 ; ; c = 112.63 ; ;$

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

$p = a+b+c = 105.1+55+112.63 = 272.73 ; ;$

2. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 272.73 }{ 2 } = 136.36 ; ;$

3. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 136.36 * (136.36-105.1)(136.36-55)(136.36-112.63) } ; ; T = sqrt{ 8233516.43 } = 2869.41 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2869.41 }{ 105.1 } = 54.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2869.41 }{ 55 } = 104.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2869.41 }{ 112.63 } = 50.95 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 105.1**2-55**2-112.63**2 }{ 2 * 55 * 112.63 } ) = 67° 53'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 55**2-105.1**2-112.63**2 }{ 2 * 105.1 * 112.63 } ) = 29° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 112.63**2-105.1**2-55**2 }{ 2 * 55 * 105.1 } ) = 83° 6'56" ; ;$

6. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 2869.41 }{ 136.36 } = 21.04 ; ;$

7. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 105.1 }{ 2 * sin 67° 53'4" } = 56.72 ; ;$

#2 Obtuse scalene triangle.

Sides: a = 105.1 b = 55 c = 71.2166451293

Area: T = 1814.363340814
Perimeter: p = 231.3166451293
Semiperimeter: s = 115.6588225647

Angle ∠ A = α = 112.1155455533° = 112°6'56″ = 1.95767838414 rad
Angle ∠ B = β = 29° = 0.50661454831 rad
Angle ∠ C = γ = 38.88545444666° = 38°53'4″ = 0.67986633291 rad

Height: h_a = 34.52664207067
Height: h_b = 65.97768512049
Height: h_c = 50.95334910879

Median: m_a = 35.87332346937
Median: m_b = 85.45655233287
Median: m_c = 75.94444485549

Inradius: r = 15.6877283788
Circumradius: R = 56.72332968398

Vertex coordinates: A[71.2166451293; 0] B[0; 0] C[91.92325312204; 50.95334910879]
Centroid: CG[54.38796608378; 16.98444970293]
Coordinates of the circumscribed circle: U[35.60882256465; 44.15441240506]
Coordinates of the inscribed circle: I[60.65882256465; 15.6877283788]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 67.88545444666° = 67°53'4″ = 1.95767838414 rad
∠ B' = β' = 151° = 0.50661454831 rad
∠ C' = γ' = 141.1155455533° = 141°6'56″ = 0.67986633291 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

$a = 105.1 ; ; b = 55 ; ; beta = 29° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 55**2 = 105.1**2 + c**2 -2 * 55 * c * cos (29° ) ; ; ; ; c**2 -183.845c +8021.01 =0 ; ; p=1; q=-183.845062441; r=8021.01 ; ; D = q**2 - 4pr = 183.845**2 - 4 * 1 * 8021.01 = 1714.96698383 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 183.85 ± sqrt{ 1714.97 } }{ 2 } ; ; c_{1,2} = 91.9225312204 ± 20.7060799273 ; ;$
$c_{1} = 112.628611148 ; ; c_{2} = 71.216451293 ; ; ; ; (c -112.628611148) (c -71.216451293) = 0 ; ; ; ; c>0 ; ;$

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.

$a = 105.1 ; ; b = 55 ; ; c = 71.22 ; ;$

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

$p = a+b+c = 105.1+55+71.22 = 231.32 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 231.32 }{ 2 } = 115.66 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 115.66 * (115.66-105.1)(115.66-55)(115.66-71.22) } ; ; T = sqrt{ 3291914.58 } = 1814.36 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1814.36 }{ 105.1 } = 34.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1814.36 }{ 55 } = 65.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1814.36 }{ 71.22 } = 50.95 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 105.1**2-55**2-71.22**2 }{ 2 * 55 * 71.22 } ) = 112° 6'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 55**2-105.1**2-71.22**2 }{ 2 * 105.1 * 71.22 } ) = 29° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 71.22**2-105.1**2-55**2 }{ 2 * 55 * 105.1 } ) = 38° 53'4" ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 1814.36 }{ 115.66 } = 15.69 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 105.1 }{ 2 * sin 112° 6'56" } = 56.72 ; ;$

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