Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 2.23660679775   b = 7.81102496759   c = 5.65768542495

Area: T = 2
Perimeter: p = 15.70331719029
Semiperimeter: s = 7.85215859514

Angle ∠ A = α = 5.19444289077° = 5°11'40″ = 0.09106598872 rad
Angle ∠ B = β = 161.5655051177° = 161°33'54″ = 2.82198420992 rad
Angle ∠ C = γ = 13.24105199152° = 13°14'26″ = 0.23110906672 rad

Height: ha = 1.7898854382
Height: hb = 0.51221475197
Height: hc = 0.70771067812

Median: ma = 6.72768120235
Median: mb = 1.80327756377
Median: mc = 5

Inradius: r = 0.25547256073
Circumradius: R = 12.34990890352

Vertex coordinates: A[0; 3] B[4; 7] C[5; 9]
Centroid: CG[3; 6.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.7644176822; 0.25547256073]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 174.8065571092° = 174°48'20″ = 0.09106598872 rad
∠ B' = β' = 18.43549488229° = 18°26'6″ = 2.82198420992 rad
∠ C' = γ' = 166.7599480085° = 166°45'34″ = 0.23110906672 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{ (4-5)**2 + (7-9)**2 } ; ; a = sqrt{ 5 } = 2.24 ; ;

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{ (0-5)**2 + (3-9)**2 } ; ; b = sqrt{ 61 } = 7.81 ; ;

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 + (3-7)**2 } ; ; c = sqrt{ 32 } = 5.66 ; ;


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 = 2.24 ; ; b = 7.81 ; ; c = 5.66 ; ;

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

p = a+b+c = 2.24+7.81+5.66 = 15.7 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.7 }{ 2 } = 7.85 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.85 * (7.85-2.24)(7.85-7.81)(7.85-5.66) } ; ; T = sqrt{ 4 } = 2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2 }{ 2.24 } = 1.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2 }{ 7.81 } = 0.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2 }{ 5.66 } = 0.71 ; ;

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{ 2.24**2-7.81**2-5.66**2 }{ 2 * 7.81 * 5.66 } ) = 5° 11'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.81**2-2.24**2-5.66**2 }{ 2 * 2.24 * 5.66 } ) = 161° 33'54" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.66**2-2.24**2-7.81**2 }{ 2 * 7.81 * 2.24 } ) = 13° 14'26" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2 }{ 7.85 } = 0.25 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2.24 }{ 2 * sin 5° 11'40" } = 12.35 ; ;




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