Triangle calculator VC

Please enter the coordinates of the three vertices


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: ha = 4.51986182011
Height: hb = 5.12991024856
Height: hc = 7.1188052168

Median: ma = 4.77696960071
Median: mb = 5.83109518948
Median: mc = 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 = | beta gamma | = | beta - gamma | ; ; a**2 = ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 + ( beta _z- gamma _z)**2 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 + ( beta _z- gamma _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 = | alpha gamma | = | alpha - gamma | ; ; b**2 = ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 + ( alpha _z- gamma _z)**2 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 + ( alpha _z- gamma _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 = | alpha beta | = | alpha - beta | ; ; c**2 = ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 + ( alpha _z- beta _z)**2 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 + ( alpha _z- beta _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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 8.19**2-7.21**2-5.2**2 }{ 2 * 7.21 * 5.2 } ) = 80° 47'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.21**2-8.19**2-5.2**2 }{ 2 * 8.19 * 5.2 } ) = 60° 24'47" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.2**2-8.19**2-7.21**2 }{ 2 * 7.21 * 8.19 } ) = 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 ; ;




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