Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 9.43439811321   b = 7.21111025509   c = 4.12331056256

Area: T = 14
Perimeter: p = 20.76881893086
Semiperimeter: s = 10.38440946543

Angle ∠ A = α = 109.6543824058° = 109°39'14″ = 1.91438202672 rad
Angle ∠ B = β = 46.0421626676° = 46°2'30″ = 0.80435779785 rad
Angle ∠ C = γ = 24.30545492659° = 24°18'16″ = 0.42441944079 rad

Height: ha = 2.9687994064
Height: hb = 3.88329013736
Height: hc = 6.7910997501

Median: ma = 3.5
Median: mb = 6.32545553203
Median: mc = 8.1399410298

Inradius: r = 1.34882157536
Circumradius: R = 5.00987932882

Vertex coordinates: A[1; 2; 0] B[3; 0; -3] C[5; 2; 6]
Centroid: CG[3; 1.33333333333; 1]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 70.34661759419° = 70°20'46″ = 1.91438202672 rad
∠ B' = β' = 133.9588373324° = 133°57'30″ = 0.80435779785 rad
∠ C' = γ' = 155.6955450734° = 155°41'44″ = 0.42441944079 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{ (3-5)**2 + (0-2)**2 + (-3 - 6)**2 } ; ; a = sqrt{ 89 } = 9.43 ; ;

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{ (1-5)**2 + (2-2)**2 + (0 - 6)**2 } ; ; b = sqrt{ 52 } = 7.21 ; ;

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{ (1-3)**2 + (2-0)**2 + (0 - (-3))**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 = 9.43 ; ; b = 7.21 ; ; c = 4.12 ; ;

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

p = a+b+c = 9.43+7.21+4.12 = 20.77 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.77 }{ 2 } = 10.38 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.38 * (10.38-9.43)(10.38-7.21)(10.38-4.12) } ; ; T = sqrt{ 196 } = 14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14 }{ 9.43 } = 2.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14 }{ 7.21 } = 3.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14 }{ 4.12 } = 6.79 ; ;

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{ 7.21**2+4.12**2-9.43**2 }{ 2 * 7.21 * 4.12 } ) = 109° 39'14" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9.43**2+4.12**2-7.21**2 }{ 2 * 9.43 * 4.12 } ) = 46° 2'30" ; ; gamma = 180° - alpha - beta = 180° - 109° 39'14" - 46° 2'30" = 24° 18'16" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14 }{ 10.38 } = 1.35 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 9.43 }{ 2 * sin 109° 39'14" } = 5.01 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.21**2+2 * 4.12**2 - 9.43**2 } }{ 2 } = 3.5 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.12**2+2 * 9.43**2 - 7.21**2 } }{ 2 } = 6.325 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.21**2+2 * 9.43**2 - 4.12**2 } }{ 2 } = 8.139 ; ;
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