Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 3.60655512755   b = 2.23660679775   c = 4.4722135955

Area: T = 4
Perimeter: p = 10.3143755208
Semiperimeter: s = 5.1576877604

Angle ∠ A = α = 53.13301023542° = 53°7'48″ = 0.9277295218 rad
Angle ∠ B = β = 29.74548812969° = 29°44'42″ = 0.51991461142 rad
Angle ∠ C = γ = 97.12550163489° = 97°7'30″ = 1.69551513213 rad

Height: ha = 2.21988007849
Height: hb = 3.5787708764
Height: hc = 1.7898854382

Median: ma = 3.04113812651
Median: mb = 3.9055124838
Median: mc = 2

Inradius: r = 0.77656631643
Circumradius: R = 2.25334695472

Vertex coordinates: A[0; 3] B[4; 1] C[2; 4]
Centroid: CG[2; 2.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.35774105375; 0.77656631643]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.8769897646° = 126°52'12″ = 0.9277295218 rad
∠ B' = β' = 150.2555118703° = 150°15'18″ = 0.51991461142 rad
∠ C' = γ' = 82.87549836511° = 82°52'30″ = 1.69551513213 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 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 } ; ; a = sqrt{ (4-2)**2 + (1-4)**2 } ; ; a = sqrt{ 13 } = 3.61 ; ;

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{ (0-2)**2 + (3-4)**2 } ; ; b = sqrt{ 5 } = 2.24 ; ;

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{ (0-4)**2 + (3-1)**2 } ; ; c = sqrt{ 20 } = 4.47 ; ;


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 = 3.61 ; ; b = 2.24 ; ; c = 4.47 ; ;

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

p = a+b+c = 3.61+2.24+4.47 = 10.31 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 10.31 }{ 2 } = 5.16 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.16 * (5.16-3.61)(5.16-2.24)(5.16-4.47) } ; ; T = sqrt{ 16 } = 4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4 }{ 3.61 } = 2.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4 }{ 2.24 } = 3.58 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4 }{ 4.47 } = 1.79 ; ;

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{ 2.24**2+4.47**2-3.61**2 }{ 2 * 2.24 * 4.47 } ) = 53° 7'48" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.61**2+4.47**2-2.24**2 }{ 2 * 3.61 * 4.47 } ) = 29° 44'42" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 3.61**2+2.24**2-4.47**2 }{ 2 * 3.61 * 2.24 } ) = 97° 7'30" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4 }{ 5.16 } = 0.78 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.61 }{ 2 * sin 53° 7'48" } = 2.25 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.24**2+2 * 4.47**2 - 3.61**2 } }{ 2 } = 3.041 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.47**2+2 * 3.61**2 - 2.24**2 } }{ 2 } = 3.905 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.24**2+2 * 3.61**2 - 4.47**2 } }{ 2 } = 2 ; ;
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