Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 6.70882039325   b = 5.09990195136   c = 7.28801098893

Area: T = 16.5
Perimeter: p = 19.08773333354
Semiperimeter: s = 9.54436666677

Angle ∠ A = α = 62.74546716251° = 62°44'41″ = 1.09551011079 rad
Angle ∠ B = β = 42.5110447078° = 42°30'38″ = 0.7421947268 rad
Angle ∠ C = γ = 74.74548812969° = 74°44'42″ = 1.30545442776 rad

Height: ha = 4.91993495505
Height: hb = 6.47218324596
Height: hc = 4.53328986103

Median: ma = 5.31550729064
Median: mb = 6.51992024052
Median: mc = 4.7176990566

Inradius: r = 1.72988952532
Circumradius: R = 3.77330010854

Vertex coordinates: A[-2; -2] B[0; 5] C[3; -1]
Centroid: CG[0.33333333333; 0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.8866067549; 1.72988952532]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 117.2555328375° = 117°15'19″ = 1.09551011079 rad
∠ B' = β' = 137.4989552922° = 137°29'22″ = 0.7421947268 rad
∠ C' = γ' = 105.2555118703° = 105°15'18″ = 1.30545442776 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{ (0-3)**2 + (5-(-1))**2 } ; ; a = sqrt{ 45 } = 6.71 ; ;

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{ (-2-3)**2 + (-2-(-1))**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{ (-2-0)**2 + (-2-5)**2 } ; ; c = sqrt{ 53 } = 7.28 ; ;


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 = 6.71 ; ; b = 5.1 ; ; c = 7.28 ; ;

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

p = a+b+c = 6.71+5.1+7.28 = 19.09 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 19.09 }{ 2 } = 9.54 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.54 * (9.54-6.71)(9.54-5.1)(9.54-7.28) } ; ; T = sqrt{ 272.25 } = 16.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16.5 }{ 6.71 } = 4.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16.5 }{ 5.1 } = 6.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16.5 }{ 7.28 } = 4.53 ; ;

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{ 6.71**2-5.1**2-7.28**2 }{ 2 * 5.1 * 7.28 } ) = 62° 44'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.1**2-6.71**2-7.28**2 }{ 2 * 6.71 * 7.28 } ) = 42° 30'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.28**2-6.71**2-5.1**2 }{ 2 * 5.1 * 6.71 } ) = 74° 44'42" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 16.5 }{ 9.54 } = 1.73 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.71 }{ 2 * sin 62° 44'41" } = 3.77 ; ;




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