Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 8.54440037453   b = 13.45436240471   c = 18.38547763109

Area: T = 53.5
Perimeter: p = 40.38224041032
Semiperimeter: s = 20.19112020516

Angle ∠ A = α = 25.63326524522° = 25°37'58″ = 0.44773741813 rad
Angle ∠ B = β = 42.93661802715° = 42°56'10″ = 0.7499377714 rad
Angle ∠ C = γ = 111.4311167276° = 111°25'52″ = 1.94548407583 rad

Height: ha = 12.523340275
Height: hb = 7.95332473648
Height: hc = 5.82200327375

Median: ma = 15.5322224567
Median: mb = 12.65989889012
Median: mc = 6.51992024052

Inradius: r = 2.65496688936
Circumradius: R = 9.87551862259

Vertex coordinates: A[-10; -3] B[7; 4] C[-1; 7]
Centroid: CG[-1.33333333333; 2.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.84877749791; 2.65496688936]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.3677347548° = 154°22'2″ = 0.44773741813 rad
∠ B' = β' = 137.0643819728° = 137°3'50″ = 0.7499377714 rad
∠ C' = γ' = 68.56988327238° = 68°34'8″ = 1.94548407583 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{ (7-(-1))**2 + (4-7)**2 } ; ; a = sqrt{ 73 } = 8.54 ; ;

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{ (-10-(-1))**2 + (-3-7)**2 } ; ; b = sqrt{ 181 } = 13.45 ; ;

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{ (-10-7)**2 + (-3-4)**2 } ; ; c = sqrt{ 338 } = 18.38 ; ;


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 = 8.54 ; ; b = 13.45 ; ; c = 18.38 ; ;

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

p = a+b+c = 8.54+13.45+18.38 = 40.38 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40.38 }{ 2 } = 20.19 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.19 * (20.19-8.54)(20.19-13.45)(20.19-18.38) } ; ; T = sqrt{ 2862.25 } = 53.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 53.5 }{ 8.54 } = 12.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 53.5 }{ 13.45 } = 7.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 53.5 }{ 18.38 } = 5.82 ; ;

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{ 8.54**2-13.45**2-18.38**2 }{ 2 * 13.45 * 18.38 } ) = 25° 37'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13.45**2-8.54**2-18.38**2 }{ 2 * 8.54 * 18.38 } ) = 42° 56'10" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 18.38**2-8.54**2-13.45**2 }{ 2 * 13.45 * 8.54 } ) = 111° 25'52" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 53.5 }{ 20.19 } = 2.65 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.54 }{ 2 * sin 25° 37'58" } = 9.88 ; ;




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