Triangle calculator SSA

Right scalene triangle.

Sides: a = 31 b = 96 c = 90.8577030548

Area: T = 1408.2843973494
Perimeter: p = 217.8577030548
Semiperimeter: s = 108.9298515274

Angle ∠ A = α = 18.83994045473° = 18°50'22″ = 0.32988096385 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 71.16105954527° = 71°9'38″ = 1.24219866883 rad

Height: h_a = 90.8577030548
Height: h_b = 29.33992494478
Height: h_c = 31

Median: m_a = 92.17696804812
Median: m_b = 48
Median: m_c = 54.99877272258

Inradius: r = 12.9298515274
Circumradius: R = 48

Vertex coordinates: A[90.8577030548; 0] B[0; 0] C[-0; 31]
Centroid: CG[30.28656768493; 10.33333333333]
Coordinates of the circumscribed circle: U[45.4298515274; 15.5]
Coordinates of the inscribed circle: I[12.9298515274; 12.9298515274]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.16105954527° = 161°9'38″ = 0.32988096385 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 108.83994045473° = 108°50'22″ = 1.24219866883 rad

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How did we calculate this triangle?

1. Use Law of Cosines

$a = 31 ; ; b = 96 ; ; beta = 90° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 96**2 = 31**2 + c**2 -2 * 31 * c * cos (90° ) ; ; ; ; c**2 -8255 =0 ; ; p=1; q=-0; r=-8255 ; ; D = q**2 - 4pr = 0**2 - 4 * 1 * (-8255) = 33020 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ ± sqrt{ 33020 } }{ 2 } = fraction{ ± 2 sqrt{ 8255 } }{ 2 } ; ; c_{1,2} = ± sqrt{ 8255} = ± 90.857030547999 ; ;$
$c_{1} = sqrt{ 8255} = 90.857030547999 ; ; c_{2} = - sqrt{ 8255} = -90.857030547999 ; ; ; ; text{ Factored form: } ; ; (c -90.857030547999) (c +90.857030547999) = 0 ; ; ; ; c>0 ; ;$

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.

$a = 31 ; ; b = 96 ; ; c = 90.86 ; ;$

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

$p = a+b+c = 31+96+90.86 = 217.86 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 217.86 }{ 2 } = 108.93 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 108.93 * (108.93-31)(108.93-96)(108.93-90.86) } ; ; T = sqrt{ 1983263.75 } = 1408.28 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1408.28 }{ 31 } = 90.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1408.28 }{ 96 } = 29.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1408.28 }{ 90.86 } = 31 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 96**2+90.86**2-31**2 }{ 2 * 96 * 90.86 } ) = 18° 50'22" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 31**2+90.86**2-96**2 }{ 2 * 31 * 90.86 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 18° 50'22" - 90° = 71° 9'38" ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 1408.28 }{ 108.93 } = 12.93 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 31 }{ 2 * sin 18° 50'22" } = 48 ; ;$

9. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 96**2+2 * 90.86**2 - 31**2 } }{ 2 } = 92.17 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90.86**2+2 * 31**2 - 96**2 } }{ 2 } = 48 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 96**2+2 * 31**2 - 90.86**2 } }{ 2 } = 54.998 ; ;$
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