Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 6.40331242374   b = 12.04215945788   c = 17.26326765016

Area: T = 26.5
Perimeter: p = 35.70773953179
Semiperimeter: s = 17.85436976589

Angle ∠ A = α = 14.77216214922° = 14°46'18″ = 0.25878134309 rad
Angle ∠ B = β = 28.65218284526° = 28°39'7″ = 0.55000687432 rad
Angle ∠ C = γ = 136.5776550055° = 136°34'36″ = 2.38437104795 rad

Height: ha = 8.27772093801
Height: hb = 4.40114104322
Height: hc = 3.07702075657

Median: ma = 14.53444418537
Median: mb = 11.54333963806
Median: mc = 4.30111626335

Inradius: r = 1.4844286365
Circumradius: R = 12.55767774253

Vertex coordinates: A[2; 7] B[5; -10] C[1; -5]
Centroid: CG[2.66766666667; -2.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.7176524102; 1.4844286365]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.2288378508° = 165°13'42″ = 0.25878134309 rad
∠ B' = β' = 151.3488171547° = 151°20'53″ = 0.55000687432 rad
∠ C' = γ' = 43.42334499448° = 43°25'24″ = 2.38437104795 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{ (5-1)**2 + (-10-(-5))**2 } ; ; a = sqrt{ 41 } = 6.4 ; ;

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{ (2-1)**2 + (7-(-5))**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{ (2-5)**2 + (7-(-10))**2 } ; ; c = sqrt{ 298 } = 17.26 ; ;


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 = 6.4 ; ; b = 12.04 ; ; c = 17.26 ; ;

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

p = a+b+c = 6.4+12.04+17.26 = 35.71 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 35.71 }{ 2 } = 17.85 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.85 * (17.85-6.4)(17.85-12.04)(17.85-17.26) } ; ; T = sqrt{ 702.25 } = 26.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 26.5 }{ 6.4 } = 8.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 26.5 }{ 12.04 } = 4.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 26.5 }{ 17.26 } = 3.07 ; ;

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{ 6.4**2-12.04**2-17.26**2 }{ 2 * 12.04 * 17.26 } ) = 14° 46'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.04**2-6.4**2-17.26**2 }{ 2 * 6.4 * 17.26 } ) = 28° 39'7" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 17.26**2-6.4**2-12.04**2 }{ 2 * 12.04 * 6.4 } ) = 136° 34'36" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 26.5 }{ 17.85 } = 1.48 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.4 }{ 2 * sin 14° 46'18" } = 12.56 ; ;




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