Triangle calculator SAS

Acute scalene triangle.

Sides: a = 9.9854607064 b = 13.313280942 c = 122.0000042458

Area: T = 57.55774214087
Perimeter: p = 35.29774207298
Semiperimeter: s = 17.64987103649

Angle ∠ A = α = 46.10221137474° = 46°6'8″ = 0.8054633677 rad
Angle ∠ B = β = 73.89878862526° = 73°53'52″ = 1.29897614254 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: h_a = 11.52992311535
Height: h_b = 8.64769233642
Height: h_c = 9.5932900174

Median: m_a = 11.64987082426
Median: m_b = 8.80655957435
Median: m_c = 10.12223322887

Inradius: r = 3.26112819984
Circumradius: R = 6.92882056816

Vertex coordinates: A[122.0000042458; 0] B[0; 0] C[2.7699231748; 9.5932900174]
Centroid: CG[4.92330786646; 3.19876333913]
Coordinates of the circumscribed circle: U[66.0000021229; 3.46441028408]
Coordinates of the inscribed circle: I[4.33659009449; 3.26112819984]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.8987886253° = 133°53'52″ = 0.8054633677 rad
∠ B' = β' = 106.1022113747° = 106°6'8″ = 1.29897614254 rad
∠ C' = γ' = 120° = 1.04771975512 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

$a = 9.98 ; ; b = 13.31 ; ; gamma = 60° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 9.98**2+13.31**2 - 2 * 9.98 * 13.31 * cos 60° } ; ; c = 12 ; ;$

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.

$a = 9.98 ; ; b = 13.31 ; ; c = 12 ; ;$

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

$p = a+b+c = 9.98+13.31+12 = 35.3 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 35.3 }{ 2 } = 17.65 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.65 * (17.65-9.98)(17.65-13.31)(17.65-12) } ; ; T = sqrt{ 3312.86 } = 57.56 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 57.56 }{ 9.98 } = 11.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 57.56 }{ 13.31 } = 8.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 57.56 }{ 12 } = 9.59 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 13.31**2+12**2-9.98**2 }{ 2 * 13.31 * 12 } ) = 46° 6'8" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9.98**2+12**2-13.31**2 }{ 2 * 9.98 * 12 } ) = 73° 53'52" ; ; gamma = 180° - alpha - beta = 180° - 46° 6'8" - 73° 53'52" = 60° ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 57.56 }{ 17.65 } = 3.26 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 9.98 }{ 2 * sin 46° 6'8" } = 6.93 ; ;$

9. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.31**2+2 * 12**2 - 9.98**2 } }{ 2 } = 11.649 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 9.98**2 - 13.31**2 } }{ 2 } = 8.806 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.31**2+2 * 9.98**2 - 12**2 } }{ 2 } = 10.122 ; ;$
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