Triangle calculator VC

Acute scalene triangle.

Sides: a = 8.18553527719 b = 7.21111025509 c = 5.19661524227

Area: T = 18.49332420089
Perimeter: p = 20.59326077455
Semiperimeter: s = 10.29663038728

Angle ∠ A = α = 80.78656651981° = 80°47'8″ = 1.41099758461 rad
Angle ∠ B = β = 60.41331739829° = 60°24'47″ = 1.05444087976 rad
Angle ∠ C = γ = 38.8011160819° = 38°48'4″ = 0.67772080099 rad

Height: h_a = 4.51986182011
Height: h_b = 5.12991024856
Height: h_c = 7.1188052168

Median: m_a = 4.77696960071
Median: m_b = 5.83109518948
Median: m_c = 7.26329195232

Inradius: r = 1.79661049166
Circumradius: R = 4.14661776944

Vertex coordinates: A[2; -2; 3] B[5; 1; 6] C[8; -6; 3]
Centroid: CG[5; -2.33333333333; 4]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 99.21443348019° = 99°12'52″ = 1.41099758461 rad
∠ B' = β' = 119.5876826017° = 119°35'13″ = 1.05444087976 rad
∠ C' = γ' = 141.1998839181° = 141°11'56″ = 0.67772080099 rad

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How did we calculate this triangle?

1. We compute side a from coordinates using the Pythagorean theorem

$a = |BC| = |B-C| ; ; a**2 = (B_x-C_x)**2 + (B_y-C_y)**2 + (B_z-C_z)**2 ; ; a = sqrt{ (B_x-C_x)**2 + (B_y-C_y)**2 + (B_z-C_z)**2 } ; ; a = sqrt{ (5-8)**2 + (1-(-6))**2 + (6 - 3)**2 } ; ; a = sqrt{ 67 } = 8.19 ; ;$

2. We compute side b from coordinates using the Pythagorean theorem

$b = |AC| = |A-C| ; ; b**2 = (A_x-C_x)**2 + (A_y-C_y)**2 + (A_z-C_z)**2 ; ; b = sqrt{ (A_x-C_x)**2 + (A_y-C_y)**2 + (A_z-C_z)**2 } ; ; b = sqrt{ (2-8)**2 + (-2-(-6))**2 + (3 - 3)**2 } ; ; b = sqrt{ 52 } = 7.21 ; ;$

3. We compute side c from coordinates using the Pythagorean theorem

$c = |AB| = |A-B| ; ; c**2 = (A_x-B_x)**2 + (A_y-B_y)**2 + (A_z-B_z)**2 ; ; c = sqrt{ (A_x-B_x)**2 + (A_y-B_y)**2 + (A_z-B_z)**2 } ; ; c = sqrt{ (2-5)**2 + (-2-1)**2 + (3 - 6)**2 } ; ; c = sqrt{ 27 } = 5.2 ; ;$

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.

$a = 8.19 ; ; b = 7.21 ; ; c = 5.2 ; ;$

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

$p = a+b+c = 8.19+7.21+5.2 = 20.59 ; ;$

5. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 20.59 }{ 2 } = 10.3 ; ;$

6. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.3 * (10.3-8.19)(10.3-7.21)(10.3-5.2) } ; ; T = sqrt{ 342 } = 18.49 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18.49 }{ 8.19 } = 4.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18.49 }{ 7.21 } = 5.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18.49 }{ 5.2 } = 7.12 ; ;$

8. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 7.21**2+5.2**2-8.19**2 }{ 2 * 7.21 * 5.2 } ) = 80° 47'8" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.19**2+5.2**2-7.21**2 }{ 2 * 8.19 * 5.2 } ) = 60° 24'47" ; ; gamma = 180° - alpha - beta = 180° - 80° 47'8" - 60° 24'47" = 38° 48'4" ; ;$

9. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 18.49 }{ 10.3 } = 1.8 ; ;$

10. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.19 }{ 2 * sin 80° 47'8" } = 4.15 ; ;$

11. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.21**2+2 * 5.2**2 - 8.19**2 } }{ 2 } = 4.77 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.2**2+2 * 8.19**2 - 7.21**2 } }{ 2 } = 5.831 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.21**2+2 * 8.19**2 - 5.2**2 } }{ 2 } = 7.263 ; ;$
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