Triangle calculator SAS

Right scalene triangle.

Sides: a = 45 b = 656 c = 657.5421633663

Area: T = 14760
Perimeter: p = 1358.542163366
Semiperimeter: s = 679.2710816831

Angle ∠ A = α = 3.9244203157° = 3°55'27″ = 0.06884902656 rad
Angle ∠ B = β = 86.0765796843° = 86°4'33″ = 1.50223060612 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: h_a = 656
Height: h_b = 45
Height: h_c = 44.89444956315

Median: m_a = 656.3865747865
Median: m_b = 331.0722499613
Median: m_c = 328.7710816831

Inradius: r = 21.72991831686
Circumradius: R = 328.7710816831

Vertex coordinates: A[657.5421633663; 0] B[0; 0] C[3.08796529016; 44.89444956315]
Centroid: CG[220.2077095522; 14.96548318772]
Coordinates of the circumscribed circle: U[328.7710816831; 0]
Coordinates of the inscribed circle: I[23.27108168314; 21.72991831686]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 176.0765796843° = 176°4'33″ = 0.06884902656 rad
∠ B' = β' = 93.9244203157° = 93°55'27″ = 1.50223060612 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Calculation of the third side c of the triangle using a Law of Cosines

$a = 45 ; ; b = 656 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 45**2+656**2 - 2 * 45 * 656 * cos 90° } ; ; c = 657.54 ; ;$
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.

$a = 45 ; ; b = 656 ; ; c = 657.54 ; ;$

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

$p = a+b+c = 45+656+657.54 = 1358.54 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 1358.54 }{ 2 } = 679.27 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 679.27 * (679.27-45)(679.27-656)(679.27-657.54) } ; ; T = sqrt{ 217857600 } = 14760 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14760 }{ 45 } = 656 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14760 }{ 656 } = 45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14760 }{ 657.54 } = 44.89 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 656**2+657.54**2-45**2 }{ 2 * 656 * 657.54 } ) = 3° 55'27" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 45**2+657.54**2-656**2 }{ 2 * 45 * 657.54 } ) = 86° 4'33" ; ;$
$gamma = 180° - alpha - beta = 180° - 3° 55'27" - 86° 4'33" = 90° ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 14760 }{ 679.27 } = 21.73 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 45 }{ 2 * sin 3° 55'27" } = 328.77 ; ;$

9. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 656**2+2 * 657.54**2 - 45**2 } }{ 2 } = 656.386 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 657.54**2+2 * 45**2 - 656**2 } }{ 2 } = 331.072 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 656**2+2 * 45**2 - 657.54**2 } }{ 2 } = 328.771 ; ;$
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