Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 11.66219037897   b = 13.03884048104   c = 5.83109518948

Area: T = 34
Perimeter: p = 30.53112604949
Semiperimeter: s = 15.26656302475

Angle ∠ A = α = 63.43549488229° = 63°26'6″ = 1.10771487178 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 26.56550511771° = 26°33'54″ = 0.4643647609 rad

Height: ha = 5.83109518948
Height: hb = 5.21553619242
Height: hc = 11.66219037897

Median: ma = 8.24662112512
Median: mb = 6.51992024052
Median: mc = 12.02108152802

Inradius: r = 2.22772254371
Circumradius: R = 6.51992024052

Vertex coordinates: A[5; 3] B[0; 6] C[-6; -4]
Centroid: CG[-0.33333333333; 1.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 2.22772254371]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.5655051177° = 116°33'54″ = 1.10771487178 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 153.4354948823° = 153°26'6″ = 0.4643647609 rad

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How did we calculate this triangle?

1. We compute side a from coordinates using the Pythagorean theorem

a = |BC| = |B-C| ; ; a**2 = (B_x-C_x)**2 + (B_y-C_y)**2 ; ; a = sqrt{ (B_x-C_x)**2 + (B_y-C_y)**2 } ; ; a = sqrt{ (0-(-6))**2 + (6-(-4))**2 } ; ; a = sqrt{ 136 } = 11.66 ; ;

2. We compute side b from coordinates using the Pythagorean theorem

b = |AC| = |A-C| ; ; b**2 = (A_x-C_x)**2 + (A_y-C_y)**2 ; ; b = sqrt{ (A_x-C_x)**2 + (A_y-C_y)**2 } ; ; b = sqrt{ (5-(-6))**2 + (3-(-4))**2 } ; ; b = sqrt{ 170 } = 13.04 ; ;

3. We compute side c from coordinates using the Pythagorean theorem

c = |AB| = |A-B| ; ; c**2 = (A_x-B_x)**2 + (A_y-B_y)**2 ; ; c = sqrt{ (A_x-B_x)**2 + (A_y-B_y)**2 } ; ; c = sqrt{ (5-0)**2 + (3-6)**2 } ; ; c = sqrt{ 34 } = 5.83 ; ;
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 = 11.66 ; ; b = 13.04 ; ; c = 5.83 ; ;

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

p = a+b+c = 11.66+13.04+5.83 = 30.53 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30.53 }{ 2 } = 15.27 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.27 * (15.27-11.66)(15.27-13.04)(15.27-5.83) } ; ; T = sqrt{ 1156 } = 34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 34 }{ 11.66 } = 5.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 34 }{ 13.04 } = 5.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 34 }{ 5.83 } = 11.66 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 13.04**2+5.83**2-11.66**2 }{ 2 * 13.04 * 5.83 } ) = 63° 26'6" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 11.66**2+5.83**2-13.04**2 }{ 2 * 11.66 * 5.83 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 63° 26'6" - 90° = 26° 33'54" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 34 }{ 15.27 } = 2.23 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 11.66 }{ 2 * sin 63° 26'6" } = 6.52 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.04**2+2 * 5.83**2 - 11.66**2 } }{ 2 } = 8.246 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.83**2+2 * 11.66**2 - 13.04**2 } }{ 2 } = 6.519 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.04**2+2 * 11.66**2 - 5.83**2 } }{ 2 } = 12.021 ; ;
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