Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 9.84988578018   b = 12.16655250606   c = 10.63301458127

Area: T = 50
Perimeter: p = 32.64545286751
Semiperimeter: s = 16.32222643376

Angle ∠ A = α = 50.64882473737° = 50°38'54″ = 0.8843978677 rad
Angle ∠ B = β = 72.77765638089° = 72°46'36″ = 1.27701906568 rad
Angle ∠ C = γ = 56.57551888174° = 56°34'31″ = 0.98774233198 rad

Height: ha = 10.15334616513
Height: hb = 8.22199493653
Height: hc = 9.40772086838

Median: ma = 10.3087764064
Median: mb = 8.24662112512
Median: mc = 9.70882439195

Inradius: r = 3.0633300469
Circumradius: R = 6.36883357324

Vertex coordinates: A[-7; -1] B[1; 6] C[5; -3]
Centroid: CG[-0.33333333333; 0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.95496231454; 3.0633300469]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.3521752626° = 129°21'6″ = 0.8843978677 rad
∠ B' = β' = 107.2233436191° = 107°13'24″ = 1.27701906568 rad
∠ C' = γ' = 123.4254811183° = 123°25'29″ = 0.98774233198 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{ (1-5)**2 + (6-(-3))**2 } ; ; a = sqrt{ 97 } = 9.85 ; ;

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{ (-7-5)**2 + (-1-(-3))**2 } ; ; b = sqrt{ 148 } = 12.17 ; ;

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{ (-7-1)**2 + (-1-6)**2 } ; ; c = sqrt{ 113 } = 10.63 ; ;


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 = 9.85 ; ; b = 12.17 ; ; c = 10.63 ; ;

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

p = a+b+c = 9.85+12.17+10.63 = 32.64 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 32.64 }{ 2 } = 16.32 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.32 * (16.32-9.85)(16.32-12.17)(16.32-10.63) } ; ; T = sqrt{ 2500 } = 50 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 50 }{ 9.85 } = 10.15 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 50 }{ 12.17 } = 8.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 50 }{ 10.63 } = 9.41 ; ;

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{ 9.85**2-12.17**2-10.63**2 }{ 2 * 12.17 * 10.63 } ) = 50° 38'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.17**2-9.85**2-10.63**2 }{ 2 * 9.85 * 10.63 } ) = 72° 46'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.63**2-9.85**2-12.17**2 }{ 2 * 12.17 * 9.85 } ) = 56° 34'31" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 50 }{ 16.32 } = 3.06 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.85 }{ 2 * sin 50° 38'54" } = 6.37 ; ;




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