Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 5.83109518948   b = 8.06222577483   c = 3.60655512755

Area: T = 9.5
Perimeter: p = 17.49987609186
Semiperimeter: s = 8.74993804593

Angle ∠ A = α = 40.81550838749° = 40°48'54″ = 0.71223575981 rad
Angle ∠ B = β = 115.3466175942° = 115°20'46″ = 2.01331705498 rad
Angle ∠ C = γ = 23.83987401832° = 23°50'19″ = 0.41660645057 rad

Height: ha = 3.25884731177
Height: hb = 2.35766599572
Height: hc = 5.27696518641

Median: ma = 5.52326805086
Median: mb = 2.69325824036
Median: mc = 6.80107352544

Inradius: r = 1.08657911648
Circumradius: R = 4.46105069088

Vertex coordinates: A[-5; 3] B[-2; 1] C[3; 4]
Centroid: CG[-1.33333333333; 2.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.51443221307; 1.08657911648]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.1854916125° = 139°11'6″ = 0.71223575981 rad
∠ B' = β' = 64.65438240581° = 64°39'14″ = 2.01331705498 rad
∠ C' = γ' = 156.1611259817° = 156°9'41″ = 0.41660645057 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{ (-2-3)**2 + (1-4)**2 } ; ; a = sqrt{ 34 } = 5.83 ; ;

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{ (-5-3)**2 + (3-4)**2 } ; ; b = sqrt{ 65 } = 8.06 ; ;

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{ (-5-(-2))**2 + (3-1)**2 } ; ; c = sqrt{ 13 } = 3.61 ; ;


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 = 5.83 ; ; b = 8.06 ; ; c = 3.61 ; ;

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

p = a+b+c = 5.83+8.06+3.61 = 17.5 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.5 }{ 2 } = 8.75 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.75 * (8.75-5.83)(8.75-8.06)(8.75-3.61) } ; ; T = sqrt{ 90.25 } = 9.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9.5 }{ 5.83 } = 3.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9.5 }{ 8.06 } = 2.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9.5 }{ 3.61 } = 5.27 ; ;

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{ 5.83**2-8.06**2-3.61**2 }{ 2 * 8.06 * 3.61 } ) = 40° 48'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.06**2-5.83**2-3.61**2 }{ 2 * 5.83 * 3.61 } ) = 115° 20'46" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.61**2-5.83**2-8.06**2 }{ 2 * 8.06 * 5.83 } ) = 23° 50'19" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9.5 }{ 8.75 } = 1.09 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.83 }{ 2 * sin 40° 48'54" } = 4.46 ; ;




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