Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 5.65768542495   b = 2.23660679775   c = 3.60655512755

Area: T = 2
Perimeter: p = 11.49884735025
Semiperimeter: s = 5.74992367512

Angle ∠ A = α = 150.2555118703° = 150°15'18″ = 2.62224465393 rad
Angle ∠ B = β = 11.3109932474° = 11°18'36″ = 0.19773955598 rad
Angle ∠ C = γ = 18.43549488229° = 18°26'6″ = 0.32217505544 rad

Height: ha = 0.70771067812
Height: hb = 1.7898854382
Height: hc = 1.10994003925

Median: ma = 1
Median: mb = 4.61097722286
Median: mc = 3.9055124838

Inradius: r = 0.34878722631
Circumradius: R = 5.70108771255

Vertex coordinates: A[0; 0] B[-2; 3] C[2; -1]
Centroid: CG[0; 0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.73993613157; 0.34878722631]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.74548812969° = 29°44'42″ = 2.62224465393 rad
∠ B' = β' = 168.6990067526° = 168°41'24″ = 0.19773955598 rad
∠ C' = γ' = 161.5655051177° = 161°33'54″ = 0.32217505544 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{ (-2-2)**2 + (3-(-1))**2 } ; ; a = sqrt{ 32 } = 5.66 ; ;

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{ (0-2)**2 + (0-(-1))**2 } ; ; b = sqrt{ 5 } = 2.24 ; ;

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{ (0-(-2))**2 + (0-3)**2 } ; ; c = sqrt{ 13 } = 3.61 ; ;
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 = 5.66 ; ; b = 2.24 ; ; c = 3.61 ; ;

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

p = a+b+c = 5.66+2.24+3.61 = 11.5 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.5 }{ 2 } = 5.75 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.75 * (5.75-5.66)(5.75-2.24)(5.75-3.61) } ; ; T = sqrt{ 4 } = 2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2 }{ 5.66 } = 0.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2 }{ 2.24 } = 1.79 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2 }{ 3.61 } = 1.11 ; ;

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{ 2.24**2+3.61**2-5.66**2 }{ 2 * 2.24 * 3.61 } ) = 150° 15'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.66**2+3.61**2-2.24**2 }{ 2 * 5.66 * 3.61 } ) = 11° 18'36" ; ;
gamma = 180° - alpha - beta = 180° - 150° 15'18" - 11° 18'36" = 18° 26'6" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2 }{ 5.75 } = 0.35 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.66 }{ 2 * sin 150° 15'18" } = 5.7 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.24**2+2 * 3.61**2 - 5.66**2 } }{ 2 } = 1 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.61**2+2 * 5.66**2 - 2.24**2 } }{ 2 } = 4.61 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.24**2+2 * 5.66**2 - 3.61**2 } }{ 2 } = 3.905 ; ;
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