Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 5.38551648071   b = 7.07110678119   c = 6.40331242374

Area: T = 16.5
Perimeter: p = 18.85993568564
Semiperimeter: s = 9.43296784282

Angle ∠ A = α = 46.79899106082° = 46°47'24″ = 0.81766379968 rad
Angle ∠ B = β = 73.14216012323° = 73°8'30″ = 1.27765617617 rad
Angle ∠ C = γ = 60.06884881595° = 60°4'7″ = 1.04883928951 rad

Height: ha = 6.12879461598
Height: hb = 4.66769047558
Height: hc = 5.15437341423

Median: ma = 6.18546584384
Median: mb = 4.74334164903
Median: mc = 5.40883269132

Inradius: r = 1.75497945583
Circumradius: R = 3.6944298588

Vertex coordinates: A[1; 3] B[5; -2] C[0; -4]
Centroid: CG[2; -1]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.53302407752; 1.75497945583]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.2110089392° = 133°12'36″ = 0.81766379968 rad
∠ B' = β' = 106.8588398768° = 106°51'30″ = 1.27765617617 rad
∠ C' = γ' = 119.9321511841° = 119°55'53″ = 1.04883928951 rad

Calculate another triangle




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 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 } ; ; a = sqrt{ (5-0)**2 + (-2-(-4))**2 } ; ; a = sqrt{ 29 } = 5.39 ; ;

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 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 } ; ; b = sqrt{ (1-0)**2 + (3-(-4))**2 } ; ; b = sqrt{ 50 } = 7.07 ; ;

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 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 } ; ; c = sqrt{ (1-5)**2 + (3-(-2))**2 } ; ; c = sqrt{ 41 } = 6.4 ; ;


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 = 7.07 ; ; c = 6.4 ; ;

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

p = a+b+c = 5.39+7.07+6.4 = 18.86 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.86 }{ 2 } = 9.43 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.43 * (9.43-5.39)(9.43-7.07)(9.43-6.4) } ; ; T = sqrt{ 272.25 } = 16.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16.5 }{ 5.39 } = 6.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16.5 }{ 7.07 } = 4.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16.5 }{ 6.4 } = 5.15 ; ;

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{ 5.39**2-7.07**2-6.4**2 }{ 2 * 7.07 * 6.4 } ) = 46° 47'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.07**2-5.39**2-6.4**2 }{ 2 * 5.39 * 6.4 } ) = 73° 8'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.4**2-5.39**2-7.07**2 }{ 2 * 7.07 * 5.39 } ) = 60° 4'7" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 16.5 }{ 9.43 } = 1.75 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.39 }{ 2 * sin 46° 47'24" } = 3.69 ; ;




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