Triangle calculator SSA

Right scalene triangle.

Sides: a = 78 b = 81.6 c = 23.97699812265

Area: T = 934.8299267835
Perimeter: p = 183.5769981227
Semiperimeter: s = 91.78549906133

Angle ∠ A = α = 72.91774167566° = 72°55'3″ = 1.27326490045 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 17.08325832434° = 17°4'57″ = 0.29881473223 rad

Height: h_a = 23.97699812265
Height: h_b = 22.91224820548
Height: h_c = 78

Median: m_a = 45.77772869445
Median: m_b = 40.8
Median: m_c = 78.91553977371

Inradius: r = 10.18549906133
Circumradius: R = 40.8

Vertex coordinates: A[23.97699812265; 0] B[0; 0] C[-0; 78]
Centroid: CG[7.99899937422; 26]
Coordinates of the circumscribed circle: U[11.98549906133; 39]
Coordinates of the inscribed circle: I[10.18549906133; 10.18549906133]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.0832583243° = 107°4'57″ = 1.27326490045 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 162.9177416757° = 162°55'3″ = 0.29881473223 rad

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How did we calculate this triangle?

1. Use Law of Cosines

$a = 78 ; ; b = 81.6 ; ; beta = 90° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 81.6**2 = 78**2 + c**2 -2 * 78 * c * cos (90° ) ; ; ; ; c**2 -574.56 =0 ; ; p=1; q=-0; r=-574.56 ; ; D = q**2 - 4pr = 0**2 - 4 * 1 * (-574.56) = 2298.24 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ ± sqrt{ 2298.24 } }{ 2 } ; ;$
$c_{1,2} = ± 23.9699812265 ; ; c_{1} = 23.9699812265 ; ; c_{2} = -23.9699812265 ; ; ; ; text{ Factored form: } ; ; (c -23.9699812265) (c +23.9699812265) = 0 ; ; ; ; c>0 ; ;$
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.

$a = 78 ; ; b = 81.6 ; ; c = 23.97 ; ;$

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

$p = a+b+c = 78+81.6+23.97 = 183.57 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 183.57 }{ 2 } = 91.78 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 91.78 * (91.78-78)(91.78-81.6)(91.78-23.97) } ; ; T = sqrt{ 873905.76 } = 934.83 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 934.83 }{ 78 } = 23.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 934.83 }{ 81.6 } = 22.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 934.83 }{ 23.97 } = 78 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 81.6**2+23.97**2-78**2 }{ 2 * 81.6 * 23.97 } ) = 72° 55'3" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 78**2+23.97**2-81.6**2 }{ 2 * 78 * 23.97 } ) = 90° ; ;$
$gamma = 180° - alpha - beta = 180° - 72° 55'3" - 90° = 17° 4'57" ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 934.83 }{ 91.78 } = 10.18 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 78 }{ 2 * sin 72° 55'3" } = 40.8 ; ;$

9. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 81.6**2+2 * 23.97**2 - 78**2 } }{ 2 } = 45.777 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.97**2+2 * 78**2 - 81.6**2 } }{ 2 } = 40.8 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 81.6**2+2 * 78**2 - 23.97**2 } }{ 2 } = 78.915 ; ;$
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