Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 5.83109518948   b = 11.18803398875   c = 7.81102496759

Area: T = 21.5
Perimeter: p = 24.82215414583
Semiperimeter: s = 12.41107707291

Angle ∠ A = α = 29.50107246235° = 29°30'3″ = 0.51548847764 rad
Angle ∠ B = β = 109.2310672376° = 109°13'50″ = 1.90664348771 rad
Angle ∠ C = γ = 41.26986030008° = 41°16'7″ = 0.72202730001 rad

Height: ha = 7.37444391611
Height: hb = 3.84660369213
Height: hc = 5.50655858371

Median: ma = 9.19223881554
Median: mb = 4.03111288741
Median: mc = 8.01656097709

Inradius: r = 1.7322366222
Circumradius: R = 5.92105347061

Vertex coordinates: A[-5; 4] B[1; -1] C[6; 2]
Centroid: CG[0.66766666667; 1.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.60443137984; 1.7322366222]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.4999275377° = 150°29'57″ = 0.51548847764 rad
∠ B' = β' = 70.76993276243° = 70°46'10″ = 1.90664348771 rad
∠ C' = γ' = 138.7311396999° = 138°43'53″ = 0.72202730001 rad

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How did we calculate this triangle?

1. We compute side a from coordinates using the Pythagorean theorem

a = |BC| = |B-C| ; ; a**2 = (B_x-C_x)**2 + (B_y-C_y)**2 ; ; a = sqrt{ (B_x-C_x)**2 + (B_y-C_y)**2 } ; ; a = sqrt{ (1-6)**2 + (-1-2)**2 } ; ; a = sqrt{ 34 } = 5.83 ; ;

2. We compute side b from coordinates using the Pythagorean theorem

b = |AC| = |A-C| ; ; b**2 = (A_x-C_x)**2 + (A_y-C_y)**2 ; ; b = sqrt{ (A_x-C_x)**2 + (A_y-C_y)**2 } ; ; b = sqrt{ (-5-6)**2 + (4-2)**2 } ; ; b = sqrt{ 125 } = 11.18 ; ;

3. We compute side c from coordinates using the Pythagorean theorem

c = |AB| = |A-B| ; ; c**2 = (A_x-B_x)**2 + (A_y-B_y)**2 ; ; c = sqrt{ (A_x-B_x)**2 + (A_y-B_y)**2 } ; ; c = sqrt{ (-5-1)**2 + (4-(-1))**2 } ; ; c = sqrt{ 61 } = 7.81 ; ;
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 = 5.83 ; ; b = 11.18 ; ; c = 7.81 ; ;

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

p = a+b+c = 5.83+11.18+7.81 = 24.82 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 24.82 }{ 2 } = 12.41 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.41 * (12.41-5.83)(12.41-11.18)(12.41-7.81) } ; ; T = sqrt{ 462.25 } = 21.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21.5 }{ 5.83 } = 7.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21.5 }{ 11.18 } = 3.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21.5 }{ 7.81 } = 5.51 ; ;

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{ 11.18**2+7.81**2-5.83**2 }{ 2 * 11.18 * 7.81 } ) = 29° 30'3" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.83**2+7.81**2-11.18**2 }{ 2 * 5.83 * 7.81 } ) = 109° 13'50" ; ;
 gamma = 180° - alpha - beta = 180° - 29° 30'3" - 109° 13'50" = 41° 16'7" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21.5 }{ 12.41 } = 1.73 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.83 }{ 2 * sin 29° 30'3" } = 5.92 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.18**2+2 * 7.81**2 - 5.83**2 } }{ 2 } = 9.192 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.81**2+2 * 5.83**2 - 11.18**2 } }{ 2 } = 4.031 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.18**2+2 * 5.83**2 - 7.81**2 } }{ 2 } = 8.016 ; ;
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