Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 9.48768329805   b = 5.09990195136   c = 10.19880390272

Area: T = 24
Perimeter: p = 24.78438915213
Semiperimeter: s = 12.39219457606

Angle ∠ A = α = 67.3880135052° = 67°22'49″ = 1.17660052071 rad
Angle ∠ B = β = 29.74548812969° = 29°44'42″ = 0.51991461142 rad
Angle ∠ C = γ = 82.87549836511° = 82°52'30″ = 1.44664413322 rad

Height: ha = 5.06596442563
Height: hb = 9.41435744866
Height: hc = 4.70767872433

Median: ma = 6.51992024052
Median: mb = 9.51331487952
Median: mc = 5.65768542495

Inradius: r = 1.93767418534
Circumradius: R = 5.13987011978

Vertex coordinates: A[-9; -5] B[-7; 5] C[-4; -4]
Centroid: CG[-6.66766666667; -1.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[3.38992982435; 1.93767418534]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.6219864948° = 112°37'11″ = 1.17660052071 rad
∠ B' = β' = 150.2555118703° = 150°15'18″ = 0.51991461142 rad
∠ C' = γ' = 97.12550163489° = 97°7'30″ = 1.44664413322 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-(-4))**2 + (5-(-4))**2 } ; ; a = sqrt{ 90 } = 9.49 ; ;

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{ (-9-(-4))**2 + (-5-(-4))**2 } ; ; b = sqrt{ 26 } = 5.1 ; ;

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{ (-9-(-7))**2 + (-5-5)**2 } ; ; c = sqrt{ 104 } = 10.2 ; ;


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 = 9.49 ; ; b = 5.1 ; ; c = 10.2 ; ;

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

p = a+b+c = 9.49+5.1+10.2 = 24.78 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 24.78 }{ 2 } = 12.39 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.39 * (12.39-9.49)(12.39-5.1)(12.39-10.2) } ; ; T = sqrt{ 576 } = 24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 24 }{ 9.49 } = 5.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 24 }{ 5.1 } = 9.41 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 24 }{ 10.2 } = 4.71 ; ;

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{ 9.49**2-5.1**2-10.2**2 }{ 2 * 5.1 * 10.2 } ) = 67° 22'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.1**2-9.49**2-10.2**2 }{ 2 * 9.49 * 10.2 } ) = 29° 44'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.2**2-9.49**2-5.1**2 }{ 2 * 5.1 * 9.49 } ) = 82° 52'30" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 24 }{ 12.39 } = 1.94 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.49 }{ 2 * sin 67° 22'49" } = 5.14 ; ;




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