Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 3.16222776602   b = 3.60655512755   c = 2.23660679775

Area: T = 3.5
Perimeter: p = 9.00438969131
Semiperimeter: s = 4.50219484566

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 = 2.21435943621
Height: hb = 1.94114506868
Height: hc = 3.13304951685

Median: ma = 2.55495097568
Median: mb = 2.06215528128
Median: mc = 3.20215621187

Inradius: r = 0.77774411533
Circumradius: R = 1.82110783977

Vertex coordinates: A[-1; -1] B[-2; 1] C[1; 2]
Centroid: CG[-0.66766666667; 0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.11110630219; 0.77774411533]

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{ (-2-1)**2 + (1-2)**2 } ; ; a = sqrt{ 10 } = 3.16 ; ;

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{ (-1-1)**2 + (-1-2)**2 } ; ; b = sqrt{ 13 } = 3.61 ; ;

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-(-2))**2 + (-1-1)**2 } ; ; c = sqrt{ 5 } = 2.24 ; ;


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 = 3.16 ; ; b = 3.61 ; ; c = 2.24 ; ;

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

p = a+b+c = 3.16+3.61+2.24 = 9 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 9 }{ 2 } = 4.5 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.5 * (4.5-3.16)(4.5-3.61)(4.5-2.24) } ; ; T = sqrt{ 12.25 } = 3.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.5 }{ 3.16 } = 2.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.5 }{ 3.61 } = 1.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.5 }{ 2.24 } = 3.13 ; ;

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{ 3.16**2-3.61**2-2.24**2 }{ 2 * 3.61 * 2.24 } ) = 60° 15'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.61**2-3.16**2-2.24**2 }{ 2 * 3.16 * 2.24 } ) = 81° 52'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.24**2-3.16**2-3.61**2 }{ 2 * 3.61 * 3.16 } ) = 37° 52'30" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.5 }{ 4.5 } = 0.78 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.16 }{ 2 * sin 60° 15'18" } = 1.82 ; ;




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