Triangle calculator SAS

Obtuse scalene triangle.

Sides: a = 1.1 b = 1.3 c = 2.10113255568

Area: T = 0.60663543888
Perimeter: p = 4.50113255568
Semiperimeter: s = 2.25106627784

Angle ∠ A = α = 26.35552492346° = 26°21'19″ = 0.46599858743 rad
Angle ∠ B = β = 31.64547507654° = 31°38'41″ = 0.55223050918 rad
Angle ∠ C = γ = 122° = 2.12993016874 rad

Height: h_a = 1.1022462525
Height: h_b = 0.93328529058
Height: h_c = 0.57771160845

Median: m_a = 1.65883981874
Median: m_b = 1.546605451
Median: m_c = 0.58883092096

Inradius: r = 0.26994114794
Circumradius: R = 1.23989188575

Vertex coordinates: A[2.10113255568; 0] B[0; 0] C[0.93664491578; 0.57771160845]
Centroid: CG[1.01325915715; 0.19223720282]
Coordinates of the circumscribed circle: U[1.05106627784; -0.65765269694]
Coordinates of the inscribed circle: I[0.95106627784; 0.26994114794]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.6454750765° = 153°38'41″ = 0.46599858743 rad
∠ B' = β' = 148.3555249235° = 148°21'19″ = 0.55223050918 rad
∠ C' = γ' = 58° = 2.12993016874 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

$a = 1.1 ; ; b = 1.3 ; ; gamma = 122° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 1.1**2+1.3**2 - 2 * 1.1 * 1.3 * cos 122° } ; ; c = 2.1 ; ;$
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.

$a = 1.1 ; ; b = 1.3 ; ; c = 2.1 ; ;$

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

$p = a+b+c = 1.1+1.3+2.1 = 4.5 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 4.5 }{ 2 } = 2.25 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.25 * (2.25-1.1)(2.25-1.3)(2.25-2.1) } ; ; T = sqrt{ 0.37 } = 0.61 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.61 }{ 1.1 } = 1.1 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.61 }{ 1.3 } = 0.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.61 }{ 2.1 } = 0.58 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 1.3**2+2.1**2-1.1**2 }{ 2 * 1.3 * 2.1 } ) = 26° 21'19" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.1**2+2.1**2-1.3**2 }{ 2 * 1.1 * 2.1 } ) = 31° 38'41" ; ; gamma = 180° - alpha - beta = 180° - 26° 21'19" - 31° 38'41" = 122° ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.61 }{ 2.25 } = 0.27 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.1 }{ 2 * sin 26° 21'19" } = 1.24 ; ;$

9. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.3**2+2 * 2.1**2 - 1.1**2 } }{ 2 } = 1.658 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.1**2+2 * 1.1**2 - 1.3**2 } }{ 2 } = 1.546 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.3**2+2 * 1.1**2 - 2.1**2 } }{ 2 } = 0.588 ; ;$
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