Triangle calculator

You have entered height h_c, angle α, angle β and angle γ.

Acute isosceles triangle.

Sides: a = 55.16988959481 b = 55.16988959481 c = 46.63107658155

Area: T = 1165.769914539
Perimeter: p = 156.9698557712
Semiperimeter: s = 78.48442788559

Angle ∠ A = α = 65° = 1.13444640138 rad
Angle ∠ B = β = 65° = 1.13444640138 rad
Angle ∠ C = γ = 50° = 0.8732664626 rad

Height: h_a = 42.26218261741
Height: h_b = 42.26218261741
Height: h_c = 50

Median: m_a = 42.99897188907
Median: m_b = 42.99897188907
Median: m_c = 50

Inradius: r = 14.85435370699
Circumradius: R = 30.43660708014

Vertex coordinates: A[46.63107658155; 0] B[0; 0] C[23.31553829077; 50]
Centroid: CG[23.31553829077; 16.66766666667]
Coordinates of the circumscribed circle: U[23.31553829077; 19.56439291987]
Coordinates of the inscribed circle: I[23.31553829077; 14.85435370699]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115° = 1.13444640138 rad
∠ B' = β' = 115° = 1.13444640138 rad
∠ C' = γ' = 130° = 0.8732664626 rad

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How did we calculate this triangle?

1. Input data entered: angle α, angle β, angle γ and height h_c.

$alpha = 65° ; ; beta = 65° ; ; gamma = 50° ; ; h_c = 50 ; ;$

2. From angle β, side a and side b we calculate side c - by using the law of cosines and quadratic equation:

$b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 55.169**2 = 55.169**2 + c**2 - 2 * 55.169 * c * cos 65° ; ; ; ; ; ; c**2 -46.631c =0 ; ; p=1; q=-46.631; r=0 ; ; D = q**2 - 4pr = 46.631**2 - 4 * 1 * 0 = 2174.42832054 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 46.63 ± sqrt{ 2174.43 } }{ 2 } ; ; c_{1,2} = 23.31538291 ± 23.3153829077 ; ; c_{1} = 46.6307658177 ; ; c_{2} = 2.25007212862E-9 ; ; ; ; text{ Factored form: } ; ; (c -46.6307658177) (c -2.25007212862E-9) = 0 ; ; ; ; c > 0 ; ; ; ; c = 46.631 ; ;$

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.

$a = 55.17 ; ; b = 55.17 ; ; c = 46.63 ; ;$

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

$p = a+b+c = 55.17+55.17+46.63 = 156.97 ; ;$

4. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 156.97 }{ 2 } = 78.48 ; ;$

5. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 78.48 * (78.48-55.17)(78.48-55.17)(78.48-46.63) } ; ; T = sqrt{ 1359017.7 } = 1165.77 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1165.77 }{ 55.17 } = 42.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1165.77 }{ 55.17 } = 42.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1165.77 }{ 46.63 } = 50 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 55.17**2+46.63**2-55.17**2 }{ 2 * 55.17 * 46.63 } ) = 65° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 55.17**2+46.63**2-55.17**2 }{ 2 * 55.17 * 46.63 } ) = 65° ; ; gamma = 180° - alpha - beta = 180° - 65° - 65° = 50° ; ;$

8. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 1165.77 }{ 78.48 } = 14.85 ; ;$

9. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 55.17 }{ 2 * sin 65° } = 30.44 ; ;$

10. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 55.17**2+2 * 46.63**2 - 55.17**2 } }{ 2 } = 42.99 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 46.63**2+2 * 55.17**2 - 55.17**2 } }{ 2 } = 42.99 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 55.17**2+2 * 55.17**2 - 46.63**2 } }{ 2 } = 50 ; ;$
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