Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 5.09990195136   b = 5.83109518948   c = 10.77703296143

Area: T = 5
Perimeter: p = 21.77003010227
Semiperimeter: s = 10.85501505114

Angle ∠ A = α = 9.16223470457° = 9°9'44″ = 0.16599131232 rad
Angle ∠ B = β = 10.49114770123° = 10°29'29″ = 0.18331108173 rad
Angle ∠ C = γ = 160.3466175942° = 160°20'46″ = 2.79985687132 rad

Height: ha = 1.96111613514
Height: hb = 1.71549858514
Height: hc = 0.92884766909

Median: ma = 8.27664726786
Median: mb = 7.90656941504
Median: mc = 1

Inradius: r = 0.46108231005
Circumradius: R = 16.01112460477

Vertex coordinates: A[0; -2] B[-4; 8] C[-3; 3]
Centroid: CG[-2.33333333333; 3]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.48884447429; 0.46108231005]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170.8387652954° = 170°50'16″ = 0.16599131232 rad
∠ B' = β' = 169.5098522988° = 169°30'31″ = 0.18331108173 rad
∠ C' = γ' = 19.65438240581° = 19°39'14″ = 2.79985687132 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{ (-4-(-3))**2 + (8-3)**2 } ; ; a = sqrt{ 26 } = 5.1 ; ;

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-(-3))**2 + (-2-3)**2 } ; ; b = sqrt{ 34 } = 5.83 ; ;

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-(-4))**2 + (-2-8)**2 } ; ; c = sqrt{ 116 } = 10.77 ; ;


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 = 5.1 ; ; b = 5.83 ; ; c = 10.77 ; ;

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

p = a+b+c = 5.1+5.83+10.77 = 21.7 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.7 }{ 2 } = 10.85 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.85 * (10.85-5.1)(10.85-5.83)(10.85-10.77) } ; ; T = sqrt{ 25 } = 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 }{ 5.1 } = 1.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5 }{ 5.83 } = 1.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5 }{ 10.77 } = 0.93 ; ;

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{ 5.83**2+10.77**2-5.1**2 }{ 2 * 5.83 * 10.77 } ) = 9° 9'44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.1**2+10.77**2-5.83**2 }{ 2 * 5.1 * 10.77 } ) = 10° 29'29" ; ; gamma = 180° - alpha - beta = 180° - 9° 9'44" - 10° 29'29" = 160° 20'46" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5 }{ 10.85 } = 0.46 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.1 }{ 2 * sin 9° 9'44" } = 16.01 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.83**2+2 * 10.77**2 - 5.1**2 } }{ 2 } = 8.276 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.77**2+2 * 5.1**2 - 5.83**2 } }{ 2 } = 7.906 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.83**2+2 * 5.1**2 - 10.77**2 } }{ 2 } = 1 ; ;
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