Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 6.08327625303   b = 5.09990195136   c = 2.23660679775

Area: T = 5.5
Perimeter: p = 13.41878500214
Semiperimeter: s = 6.70989250107

Angle ∠ A = α = 105.2555118703° = 105°15'18″ = 1.83770483759 rad
Angle ∠ B = β = 53.97326266149° = 53°58'21″ = 0.94220000404 rad
Angle ∠ C = γ = 20.7722254682° = 20°46'20″ = 0.36325442373 rad

Height: ha = 1.80883888604
Height: hb = 2.15772774865
Height: hc = 4.91993495505

Median: ma = 2.5
Median: mb = 3.80878865529
Median: mc = 5.5

Inradius: r = 0.82198034694
Circumradius: R = 3.1522461979

Vertex coordinates: A[0; -4] B[-1; -2] C[5; -3]
Centroid: CG[1.33333333333; -3]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.5966220705; 0.82198034694]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 74.74548812969° = 74°44'42″ = 1.83770483759 rad
∠ B' = β' = 126.0277373385° = 126°1'39″ = 0.94220000404 rad
∠ C' = γ' = 159.2287745318° = 159°13'40″ = 0.36325442373 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{ (-1-5)**2 + (-2-(-3))**2 } ; ; a = sqrt{ 37 } = 6.08 ; ;

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{ (0-5)**2 + (-4-(-3))**2 } ; ; b = sqrt{ 26 } = 5.1 ; ;

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-(-1))**2 + (-4-(-2))**2 } ; ; c = sqrt{ 5 } = 2.24 ; ;
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 = 6.08 ; ; b = 5.1 ; ; c = 2.24 ; ;

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

p = a+b+c = 6.08+5.1+2.24 = 13.42 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 13.42 }{ 2 } = 6.71 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.71 * (6.71-6.08)(6.71-5.1)(6.71-2.24) } ; ; T = sqrt{ 30.25 } = 5.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5.5 }{ 6.08 } = 1.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5.5 }{ 5.1 } = 2.16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5.5 }{ 2.24 } = 4.92 ; ;

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{ 5.1**2+2.24**2-6.08**2 }{ 2 * 5.1 * 2.24 } ) = 105° 15'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.08**2+2.24**2-5.1**2 }{ 2 * 6.08 * 2.24 } ) = 53° 58'21" ; ;
 gamma = 180° - alpha - beta = 180° - 105° 15'18" - 53° 58'21" = 20° 46'20" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5.5 }{ 6.71 } = 0.82 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.08 }{ 2 * sin 105° 15'18" } = 3.15 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.1**2+2 * 2.24**2 - 6.08**2 } }{ 2 } = 2.5 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.24**2+2 * 6.08**2 - 5.1**2 } }{ 2 } = 3.808 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.1**2+2 * 6.08**2 - 2.24**2 } }{ 2 } = 5.5 ; ;
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