Triangle calculator - result

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, b and height hc.

Obtuse scalene triangle.

Sides: a = 3   b = 2   c = 4.79106898287

Area: T = 1.677674144
Perimeter: p = 9.79106898287
Semiperimeter: s = 4.89553449143

Angle ∠ A = α = 20.48773151147° = 20°29'14″ = 0.35875711036 rad
Angle ∠ B = β = 13.49333988216° = 13°29'36″ = 0.23655042367 rad
Angle ∠ C = γ = 146.0199286064° = 146°1'9″ = 2.54985173132 rad

Height: ha = 1.11878276267
Height: hb = 1.677674144
Height: hc = 0.7

Median: ma = 3.35504260203
Median: mb = 3.87698003201
Median: mc = 0.87331109559

Inradius: r = 0.34325175283
Circumradius: R = 4.28657142857

Vertex coordinates: A[4.79106898287; 0] B[0; 0] C[2.91771904292; 0.7]
Centroid: CG[2.56992934193; 0.23333333333]
Coordinates of the circumscribed circle: U[2.39553449143; -3.55438246552]
Coordinates of the inscribed circle: I[2.89553449143; 0.34325175283]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.5132684885° = 159°30'46″ = 0.35875711036 rad
∠ B' = β' = 166.5076601178° = 166°30'24″ = 0.23655042367 rad
∠ C' = γ' = 33.98107139363° = 33°58'51″ = 2.54985173132 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and height hc.

a = 3 ; ; b = 2 ; ; h_c = 0.7 ; ;

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

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 3**2 = 2**2 + c**2 - 2 * 2 * c * cos 20° 29'14" ; ; ; ; ; ; c**2 -3.747c -5 =0 ; ; a=1; b=-3.747; c=-5 ; ; D = b**2 - 4ac = 3.747**2 - 4 * 1 * (-5) = 34.04 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 3.75 ± sqrt{ 34.04 } }{ 2 } ; ; c_{1,2} = 1.8734994 ± 2.91719042916 ; ; c_{1} = 4.79068982916 ; ; c_{2} = -1.04369102916 ; ; ; ; text{ Factored form: } ; ;
(c -4.79068982916) (c +1.04369102916) = 0 ; ; ; ; c > 0 ; ; ; ; c = 4.791 ; ;


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 ; ; b = 2 ; ; c = 4.79 ; ;

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

p = a+b+c = 3+2+4.79 = 9.79 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 9.79 }{ 2 } = 4.9 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.9 * (4.9-3)(4.9-2)(4.9-4.79) } ; ; T = sqrt{ 2.81 } = 1.68 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.68 }{ 3 } = 1.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.68 }{ 2 } = 1.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.68 }{ 4.79 } = 0.7 ; ;

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{ 2**2+4.79**2-3**2 }{ 2 * 2 * 4.79 } ) = 20° 29'14" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3**2+4.79**2-2**2 }{ 2 * 3 * 4.79 } ) = 13° 29'36" ; ; gamma = 180° - alpha - beta = 180° - 20° 29'14" - 13° 29'36" = 146° 1'9" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.68 }{ 4.9 } = 0.34 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3 }{ 2 * sin 20° 29'14" } = 4.29 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2**2+2 * 4.79**2 - 3**2 } }{ 2 } = 3.35 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.79**2+2 * 3**2 - 2**2 } }{ 2 } = 3.87 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2**2+2 * 3**2 - 4.79**2 } }{ 2 } = 0.873 ; ;
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