Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 7.61657731059   b = 12.04215945788   c = 6.40331242374

Area: T = 21.5
Perimeter: p = 26.06604919221
Semiperimeter: s = 13.0330245961

Angle ∠ A = α = 33.89661665634° = 33°53'46″ = 0.59215997103 rad
Angle ∠ B = β = 118.1421601232° = 118°8'30″ = 2.06219599251 rad
Angle ∠ C = γ = 27.96222322044° = 27°57'44″ = 0.48880330182 rad

Height: ha = 5.6466176613
Height: hb = 3.57109556337
Height: hc = 6.71554717612

Median: ma = 8.86600225733
Median: mb = 3.64400549446
Median: mc = 9.55224865873

Inradius: r = 1.65500072266
Circumradius: R = 6.82879679675

Vertex coordinates: A[1; -6] B[5; -1] C[2; 6]
Centroid: CG[2.66766666667; -0.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.88325620049; 1.65500072266]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.1043833437° = 146°6'14″ = 0.59215997103 rad
∠ B' = β' = 61.85883987677° = 61°51'30″ = 2.06219599251 rad
∠ C' = γ' = 152.0387767796° = 152°2'16″ = 0.48880330182 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{ (5-2)**2 + (-1-6)**2 } ; ; a = sqrt{ 58 } = 7.62 ; ;

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{ (1-2)**2 + (-6-6)**2 } ; ; b = sqrt{ 145 } = 12.04 ; ;

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{ (1-5)**2 + (-6-(-1))**2 } ; ; c = sqrt{ 41 } = 6.4 ; ;


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 = 7.62 ; ; b = 12.04 ; ; c = 6.4 ; ;

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

p = a+b+c = 7.62+12.04+6.4 = 26.06 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.06 }{ 2 } = 13.03 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.03 * (13.03-7.62)(13.03-12.04)(13.03-6.4) } ; ; T = sqrt{ 462.25 } = 21.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21.5 }{ 7.62 } = 5.65 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21.5 }{ 12.04 } = 3.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21.5 }{ 6.4 } = 6.72 ; ;

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{ 7.62**2-12.04**2-6.4**2 }{ 2 * 12.04 * 6.4 } ) = 33° 53'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.04**2-7.62**2-6.4**2 }{ 2 * 7.62 * 6.4 } ) = 118° 8'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.4**2-7.62**2-12.04**2 }{ 2 * 12.04 * 7.62 } ) = 27° 57'44" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21.5 }{ 13.03 } = 1.65 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.62 }{ 2 * sin 33° 53'46" } = 6.83 ; ;




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