Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 8.06222577483   b = 10.77703296143   c = 8.54440037453

Area: T = 34
Perimeter: p = 27.37765911079
Semiperimeter: s = 13.68882955539

Angle ∠ A = α = 47.64325452941° = 47°38'33″ = 0.83215192794 rad
Angle ∠ B = β = 80.81111639226° = 80°48'40″ = 1.41104208828 rad
Angle ∠ C = γ = 51.54662907833° = 51°32'47″ = 0.98996524914 rad

Height: ha = 8.43443619521
Height: hb = 6.3143641498
Height: hc = 7.95987980093

Median: ma = 8.84659030065
Median: mb = 6.32545553203
Median: mc = 8.5

Inradius: r = 2.48438738955
Circumradius: R = 5.45551688133

Vertex coordinates: A[-4; -1] B[-1; 7] C[6; 3]
Centroid: CG[0.33333333333; 3]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.40218031302; 2.48438738955]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.3577454706° = 132°21'27″ = 0.83215192794 rad
∠ B' = β' = 99.18988360774° = 99°11'20″ = 1.41104208828 rad
∠ C' = γ' = 128.4543709217° = 128°27'13″ = 0.98996524914 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{ (-1-6)**2 + (7-3)**2 } ; ; a = sqrt{ 65 } = 8.06 ; ;

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{ (-4-6)**2 + (-1-3)**2 } ; ; b = sqrt{ 116 } = 10.77 ; ;

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{ (-4-(-1))**2 + (-1-7)**2 } ; ; c = sqrt{ 73 } = 8.54 ; ;


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 = 8.06 ; ; b = 10.77 ; ; c = 8.54 ; ;

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

p = a+b+c = 8.06+10.77+8.54 = 27.38 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.38 }{ 2 } = 13.69 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.69 * (13.69-8.06)(13.69-10.77)(13.69-8.54) } ; ; T = sqrt{ 1156 } = 34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 34 }{ 8.06 } = 8.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 34 }{ 10.77 } = 6.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 34 }{ 8.54 } = 7.96 ; ;

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{ 8.06**2-10.77**2-8.54**2 }{ 2 * 10.77 * 8.54 } ) = 47° 38'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10.77**2-8.06**2-8.54**2 }{ 2 * 8.06 * 8.54 } ) = 80° 48'40" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.54**2-8.06**2-10.77**2 }{ 2 * 10.77 * 8.06 } ) = 51° 32'47" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 34 }{ 13.69 } = 2.48 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.06 }{ 2 * sin 47° 38'33" } = 5.46 ; ;




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