Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 2.44994897428   b = 4.12331056256   c = 1.73220508076

Area: T = 0.70771067812
Perimeter: p = 8.3054646176
Semiperimeter: s = 4.1522323088

Angle ∠ A = α = 11.4221753659° = 11°25'18″ = 0.19993472077 rad
Angle ∠ B = β = 160.5298779366° = 160°31'44″ = 2.80217557441 rad
Angle ∠ C = γ = 8.04994669755° = 8°2'58″ = 0.14404897018 rad

Height: ha = 0.57773502692
Height: hb = 0.34329971703
Height: hc = 0.81664965809

Median: ma = 2.91554759474
Median: mb = 0.5
Median: mc = 3.27987192622

Inradius: r = 0.17702918502
Circumradius: R = 6.18546584384

Vertex coordinates: A[-1; 2; 0] B[0; 3; 1] C[1; 4; 3]
Centroid: CG[0; 3; 1.33333333333]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 168.5788246341° = 168°34'42″ = 0.19993472077 rad
∠ B' = β' = 19.47112206345° = 19°28'16″ = 2.80217557441 rad
∠ C' = γ' = 171.9510533024° = 171°57'2″ = 0.14404897018 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{ (0-1)**2 + (3-4)**2 + (1 - 3)**2 } ; ; a = sqrt{ 6 } = 2.45 ; ;

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{ (-1-1)**2 + (2-4)**2 + (0 - 3)**2 } ; ; b = sqrt{ 17 } = 4.12 ; ;

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{ (-1-0)**2 + (2-3)**2 + (0 - 1)**2 } ; ; c = sqrt{ 3 } = 1.73 ; ;
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 = 2.45 ; ; b = 4.12 ; ; c = 1.73 ; ;

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

p = a+b+c = 2.45+4.12+1.73 = 8.3 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 8.3 }{ 2 } = 4.15 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.15 * (4.15-2.45)(4.15-4.12)(4.15-1.73) } ; ; T = sqrt{ 0.5 } = 0.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.71 }{ 2.45 } = 0.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.71 }{ 4.12 } = 0.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.71 }{ 1.73 } = 0.82 ; ;

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{ 4.12**2+1.73**2-2.45**2 }{ 2 * 4.12 * 1.73 } ) = 11° 25'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.45**2+1.73**2-4.12**2 }{ 2 * 2.45 * 1.73 } ) = 160° 31'44" ; ;
 gamma = 180° - alpha - beta = 180° - 11° 25'18" - 160° 31'44" = 8° 2'58" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.71 }{ 4.15 } = 0.17 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 2.45 }{ 2 * sin 11° 25'18" } = 6.18 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.12**2+2 * 1.73**2 - 2.45**2 } }{ 2 } = 2.915 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.73**2+2 * 2.45**2 - 4.12**2 } }{ 2 } = 0.5 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.12**2+2 * 2.45**2 - 1.73**2 } }{ 2 } = 3.279 ; ;
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