Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 14.14221356237   b = 9.84988578018   c = 5.38551648071

Area: T = 19
Perimeter: p = 29.37661582327
Semiperimeter: s = 14.68880791163

Angle ∠ A = α = 134.2366101539° = 134°14'10″ = 2.34328619469 rad
Angle ∠ B = β = 29.93215118405° = 29°55'53″ = 0.52224034317 rad
Angle ∠ C = γ = 15.83223866204° = 15°49'57″ = 0.2766327275 rad

Height: ha = 2.68770057685
Height: hb = 3.85883154275
Height: hc = 7.05664228507

Median: ma = 3.60655512755
Median: mb = 9.5
Median: mc = 11.8854864324

Inradius: r = 1.2943566017
Circumradius: R = 9.86992982066

Vertex coordinates: A[1; -6] B[6; -4] C[-8; -2]
Centroid: CG[-0.33333333333; -4]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.24767199243; 1.2943566017]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 45.76438984609° = 45°45'50″ = 2.34328619469 rad
∠ B' = β' = 150.0688488159° = 150°4'7″ = 0.52224034317 rad
∠ C' = γ' = 164.168761338° = 164°10'3″ = 0.2766327275 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{ (6-(-8))**2 + (-4-(-2))**2 } ; ; a = sqrt{ 200 } = 14.14 ; ;

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{ (1-(-8))**2 + (-6-(-2))**2 } ; ; b = sqrt{ 97 } = 9.85 ; ;

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{ (1-6)**2 + (-6-(-4))**2 } ; ; c = sqrt{ 29 } = 5.39 ; ;
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 = 14.14 ; ; b = 9.85 ; ; c = 5.39 ; ;

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

p = a+b+c = 14.14+9.85+5.39 = 29.38 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29.38 }{ 2 } = 14.69 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.69 * (14.69-14.14)(14.69-9.85)(14.69-5.39) } ; ; T = sqrt{ 361 } = 19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 19 }{ 14.14 } = 2.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 19 }{ 9.85 } = 3.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 19 }{ 5.39 } = 7.06 ; ;

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{ 9.85**2+5.39**2-14.14**2 }{ 2 * 9.85 * 5.39 } ) = 134° 14'10" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 14.14**2+5.39**2-9.85**2 }{ 2 * 14.14 * 5.39 } ) = 29° 55'53" ; ;
 gamma = 180° - alpha - beta = 180° - 134° 14'10" - 29° 55'53" = 15° 49'57" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 19 }{ 14.69 } = 1.29 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 14.14 }{ 2 * sin 134° 14'10" } = 9.87 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.85**2+2 * 5.39**2 - 14.14**2 } }{ 2 } = 3.606 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.39**2+2 * 14.14**2 - 9.85**2 } }{ 2 } = 9.5 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.85**2+2 * 14.14**2 - 5.39**2 } }{ 2 } = 11.885 ; ;
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