Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 6.32545553203   b = 7.21111025509   c = 4.4722135955

Area: T = 14
Perimeter: p = 18.00877938263
Semiperimeter: s = 9.00438969131

Angle ∠ A = α = 60.25551187031° = 60°15'18″ = 1.05216502125 rad
Angle ∠ B = β = 81.87698976458° = 81°52'12″ = 1.42988992722 rad
Angle ∠ C = γ = 37.87549836511° = 37°52'30″ = 0.66110431689 rad

Height: ha = 4.42771887242
Height: hb = 3.88329013736
Height: hc = 6.2610990337

Median: ma = 5.09990195136
Median: mb = 4.12331056256
Median: mc = 6.40331242374

Inradius: r = 1.55548823065
Circumradius: R = 3.64221567954

Vertex coordinates: A[-2; 1] B[2; 3] C[4; -3]
Centroid: CG[1.33333333333; 0.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.22221260438; 1.55548823065]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 119.7454881297° = 119°44'42″ = 1.05216502125 rad
∠ B' = β' = 98.13301023542° = 98°7'48″ = 1.42988992722 rad
∠ C' = γ' = 142.1255016349° = 142°7'30″ = 0.66110431689 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{ (2-4)**2 + (3-(-3))**2 } ; ; a = sqrt{ 40 } = 6.32 ; ;

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{ (-2-4)**2 + (1-(-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 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 } ; ; c = sqrt{ (-2-2)**2 + (1-3)**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 = 6.32 ; ; b = 7.21 ; ; c = 4.47 ; ;

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

p = a+b+c = 6.32+7.21+4.47 = 18.01 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.01 }{ 2 } = 9 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9 * (9-6.32)(9-7.21)(9-4.47) } ; ; T = sqrt{ 196 } = 14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14 }{ 6.32 } = 4.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14 }{ 7.21 } = 3.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14 }{ 4.47 } = 6.26 ; ;

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{ 6.32**2-7.21**2-4.47**2 }{ 2 * 7.21 * 4.47 } ) = 60° 15'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.21**2-6.32**2-4.47**2 }{ 2 * 6.32 * 4.47 } ) = 81° 52'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.47**2-6.32**2-7.21**2 }{ 2 * 7.21 * 6.32 } ) = 37° 52'30" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14 }{ 9 } = 1.55 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.32 }{ 2 * sin 60° 15'18" } = 3.64 ; ;




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