Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 5.38551648071   b = 3.31766247904   c = 2.44994897428

Area: T = 2.73986127875
Perimeter: p = 11.15112793403
Semiperimeter: s = 5.57656396701

Angle ∠ A = α = 137.6087954285° = 137°36'29″ = 2.40217118792 rad
Angle ∠ B = β = 24.53436377129° = 24°32'1″ = 0.42881927556 rad
Angle ∠ C = γ = 17.85884080019° = 17°51'30″ = 0.31216880188 rad

Height: ha = 1.01770952554
Height: hb = 1.65114456477
Height: hc = 2.23660679775

Median: ma = 1.11880339887
Median: mb = 3.84105728739
Median: mc = 4.30111626335

Inradius: r = 0.49111746364
Circumradius: R = 3.99437451095

Vertex coordinates: A[2; 0; 0] B[3; 2; -1] C[3; -3; 1]
Centroid: CG[2.66766666667; -0.33333333333; 0]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 42.39220457148° = 42°23'31″ = 2.40217118792 rad
∠ B' = β' = 155.4666362287° = 155°27'59″ = 0.42881927556 rad
∠ C' = γ' = 162.1421591998° = 162°8'30″ = 0.31216880188 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 + (B_z-C_z)**2 ; ; a = sqrt{ (B_x-C_x)**2 + (B_y-C_y)**2 + (B_z-C_z)**2 } ; ; a = sqrt{ (3-3)**2 + (2-(-3))**2 + (-1 - 1)**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 + (A_z-C_z)**2 ; ; b = sqrt{ (A_x-C_x)**2 + (A_y-C_y)**2 + (A_z-C_z)**2 } ; ; b = sqrt{ (2-3)**2 + (0-(-3))**2 + (0 - 1)**2 } ; ; b = sqrt{ 11 } = 3.32 ; ;

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 + (A_z-B_z)**2 ; ; c = sqrt{ (A_x-B_x)**2 + (A_y-B_y)**2 + (A_z-B_z)**2 } ; ; c = sqrt{ (2-3)**2 + (0-2)**2 + (0 - (-1))**2 } ; ; c = sqrt{ 6 } = 2.45 ; ;
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 = 3.32 ; ; c = 2.45 ; ;

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

p = a+b+c = 5.39+3.32+2.45 = 11.15 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.15 }{ 2 } = 5.58 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.58 * (5.58-5.39)(5.58-3.32)(5.58-2.45) } ; ; T = sqrt{ 7.5 } = 2.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.74 }{ 5.39 } = 1.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.74 }{ 3.32 } = 1.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.74 }{ 2.45 } = 2.24 ; ;

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{ 3.32**2+2.45**2-5.39**2 }{ 2 * 3.32 * 2.45 } ) = 137° 36'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.39**2+2.45**2-3.32**2 }{ 2 * 5.39 * 2.45 } ) = 24° 32'1" ; ; gamma = 180° - alpha - beta = 180° - 137° 36'29" - 24° 32'1" = 17° 51'30" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.74 }{ 5.58 } = 0.49 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.39 }{ 2 * sin 137° 36'29" } = 3.99 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.32**2+2 * 2.45**2 - 5.39**2 } }{ 2 } = 1.118 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.45**2+2 * 5.39**2 - 3.32**2 } }{ 2 } = 3.841 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.32**2+2 * 5.39**2 - 2.45**2 } }{ 2 } = 4.301 ; ;
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