Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 5.38551648071   b = 8.48552813742   c = 4.12331056256

Area: T = 9
Perimeter: p = 17.9943551807
Semiperimeter: s = 8.99767759035

Angle ∠ A = α = 30.96437565321° = 30°57'50″ = 0.54404195003 rad
Angle ∠ B = β = 125.8387652954° = 125°50'16″ = 2.1966281367 rad
Angle ∠ C = γ = 23.19985905136° = 23°11'55″ = 0.40548917863 rad

Height: ha = 3.34325160872
Height: hb = 2.12113203436
Height: hc = 4.36656412507

Median: ma = 6.10332778079
Median: mb = 2.23660679775
Median: mc = 6.80107352544

Inradius: r = 11.0003583613
Circumradius: R = 5.23334394894

Vertex coordinates: A[2; 2] B[1; -2] C[-4; -4]
Centroid: CG[-0.33333333333; -1.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.72224810387; 11.0003583613]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.0366243468° = 149°2'10″ = 0.54404195003 rad
∠ B' = β' = 54.16223470457° = 54°9'44″ = 2.1966281367 rad
∠ C' = γ' = 156.8011409486° = 156°48'5″ = 0.40548917863 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-(-4))**2 + (-2-(-4))**2 } ; ; a = sqrt{ 29 } = 5.39 ; ;

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{ (2-(-4))**2 + (2-(-4))**2 } ; ; b = sqrt{ 72 } = 8.49 ; ;

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{ (2-1)**2 + (2-(-2))**2 } ; ; c = sqrt{ 17 } = 4.12 ; ;
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 = 5.39 ; ; b = 8.49 ; ; c = 4.12 ; ;

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

p = a+b+c = 5.39+8.49+4.12 = 17.99 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.99 }{ 2 } = 9 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9 * (9-5.39)(9-8.49)(9-4.12) } ; ; T = sqrt{ 81 } = 9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9 }{ 5.39 } = 3.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9 }{ 8.49 } = 2.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9 }{ 4.12 } = 4.37 ; ;

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{ 8.49**2+4.12**2-5.39**2 }{ 2 * 8.49 * 4.12 } ) = 30° 57'50" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.39**2+4.12**2-8.49**2 }{ 2 * 5.39 * 4.12 } ) = 125° 50'16" ; ; gamma = 180° - alpha - beta = 180° - 30° 57'50" - 125° 50'16" = 23° 11'55" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9 }{ 9 } = 1 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.39 }{ 2 * sin 30° 57'50" } = 5.23 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.49**2+2 * 4.12**2 - 5.39**2 } }{ 2 } = 6.103 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.12**2+2 * 5.39**2 - 8.49**2 } }{ 2 } = 2.236 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.49**2+2 * 5.39**2 - 4.12**2 } }{ 2 } = 6.801 ; ;
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