Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 4.24326406871   b = 1.41442135624   c = 4.4722135955

Area: T = 3
Perimeter: p = 10.12989902045
Semiperimeter: s = 5.06444951022

Angle ∠ A = α = 71.56550511771° = 71°33'54″ = 1.24990457724 rad
Angle ∠ B = β = 18.43549488229° = 18°26'6″ = 0.32217505544 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 1.41442135624
Height: hb = 4.24326406871
Height: hc = 1.34216407865

Median: ma = 2.55495097568
Median: mb = 4.30111626335
Median: mc = 2.23660679775

Inradius: r = 0.59223591472
Circumradius: R = 2.23660679775

Vertex coordinates: A[0; 3] B[4; 1] C[1; 4]
Centroid: CG[1.66766666667; 2.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.77770774417; 0.59223591472]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108.4354948823° = 108°26'6″ = 1.24990457724 rad
∠ B' = β' = 161.5655051177° = 161°33'54″ = 0.32217505544 rad
∠ C' = γ' = 90° = 1.57107963268 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{ (4-1)**2 + (1-4)**2 } ; ; a = sqrt{ 18 } = 4.24 ; ;

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{ (0-1)**2 + (3-4)**2 } ; ; b = sqrt{ 2 } = 1.41 ; ;

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{ (0-4)**2 + (3-1)**2 } ; ; c = sqrt{ 20 } = 4.47 ; ;


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 = 4.24 ; ; b = 1.41 ; ; c = 4.47 ; ;

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

p = a+b+c = 4.24+1.41+4.47 = 10.13 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 10.13 }{ 2 } = 5.06 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.06 * (5.06-4.24)(5.06-1.41)(5.06-4.47) } ; ; T = sqrt{ 9 } = 3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3 }{ 4.24 } = 1.41 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3 }{ 1.41 } = 4.24 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3 }{ 4.47 } = 1.34 ; ;

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{ 1.41**2+4.47**2-4.24**2 }{ 2 * 1.41 * 4.47 } ) = 71° 33'54" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.24**2+4.47**2-1.41**2 }{ 2 * 4.24 * 4.47 } ) = 18° 26'6" ; ; gamma = 180° - alpha - beta = 180° - 71° 33'54" - 18° 26'6" = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3 }{ 5.06 } = 0.59 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4.24 }{ 2 * sin 71° 33'54" } = 2.24 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.41**2+2 * 4.47**2 - 4.24**2 } }{ 2 } = 2.55 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.47**2+2 * 4.24**2 - 1.41**2 } }{ 2 } = 4.301 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.41**2+2 * 4.24**2 - 4.47**2 } }{ 2 } = 2.236 ; ;
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