Triangle calculator ASA

Obtuse isosceles triangle.

Sides: a = 3.77004900064 b = 3.77004900064 c = 5.5

Area: T = 6.8099305585
Perimeter: p = 12.90109800128
Semiperimeter: s = 6.45504900064

Angle ∠ A = α = 42° = 0.73330382858 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 96° = 1.67655160819 rad

Height: h_a = 3.6880218335
Height: h_b = 3.6880218335
Height: h_c = 2.47661111218

Median: m_a = 4.30767861071
Median: m_b = 4.30767861071
Median: m_c = 2.47661111218

Inradius: r = 1.05656260963
Circumradius: R = 2.76551477688

Vertex coordinates: A[5.5; 0] B[0; 0] C[2.75; 2.47661111218]
Centroid: CG[2.75; 0.82553703739]
Coordinates of the circumscribed circle: U[2.75; -0.2899036647]
Coordinates of the inscribed circle: I[2.75; 1.05656260963]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138° = 0.73330382858 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 84° = 1.67655160819 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

$alpha = 42° ; ; beta = 42° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 42° - 42° = 96° ; ;$

2. By using the law of sines, we calculate unknown side a

$c = 5.5 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 5.5 * fraction{ sin 42° }{ sin 96° } = 3.7 ; ;$

3. By using the law of sines, we calculate last unknown side b

$fraction{ b }{ c } = fraction{ sin beta }{ sin gamma } ; ; ; ; b = c * fraction{ sin beta }{ sin gamma } ; ; ; ; b = 5.5 * fraction{ sin 42° }{ sin 96° } = 3.7 ; ;$

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.

$a = 3.7 ; ; b = 3.7 ; ; c = 5.5 ; ;$

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

$p = a+b+c = 3.7+3.7+5.5 = 12.9 ; ;$

5. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 12.9 }{ 2 } = 6.45 ; ;$

6. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.45 * (6.45-3.7)(6.45-3.7)(6.45-5.5) } ; ; T = sqrt{ 46.37 } = 6.81 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6.81 }{ 3.7 } = 3.68 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6.81 }{ 3.7 } = 3.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6.81 }{ 5.5 } = 2.48 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 3.7**2+5.5**2-3.7**2 }{ 2 * 3.7 * 5.5 } ) = 42° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.7**2+5.5**2-3.7**2 }{ 2 * 3.7 * 5.5 } ) = 42° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 3.7**2+3.7**2-5.5**2 }{ 2 * 3.7 * 3.7 } ) = 96° ; ;$

9. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 6.81 }{ 6.45 } = 1.06 ; ;$

10. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.7 }{ 2 * sin 42° } = 2.77 ; ;$

11. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.7**2+2 * 5.5**2 - 3.7**2 } }{ 2 } = 4.307 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.5**2+2 * 3.7**2 - 3.7**2 } }{ 2 } = 4.307 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.7**2+2 * 3.7**2 - 5.5**2 } }{ 2 } = 2.476 ; ;$
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