Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 4.24326406871   b = 9.48768329805   c = 8.48552813742

Area: T = 18
Perimeter: p = 22.21547550419
Semiperimeter: s = 11.10773775209

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

Height: ha = 8.48552813742
Height: hb = 3.79547331922
Height: hc = 4.24326406871

Median: ma = 8.74664278423
Median: mb = 4.74334164903
Median: mc = 6

Inradius: r = 1.62105445404
Circumradius: R = 4.74334164903

Vertex coordinates: A[0; 0] B[6; 6] C[9; 3]
Centroid: CG[5; 3]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 1.62105445404]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.4354948823° = 153°26'6″ = 0.4643647609 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 116.5655051177° = 116°33'54″ = 1.10771487178 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{ (6-9)**2 + (6-3)**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{ (0-9)**2 + (0-3)**2 } ; ; b = sqrt{ 90 } = 9.49 ; ;

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{ (0-6)**2 + (0-6)**2 } ; ; c = sqrt{ 72 } = 8.49 ; ;
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 = 9.49 ; ; c = 8.49 ; ;

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

p = a+b+c = 4.24+9.49+8.49 = 22.21 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.21 }{ 2 } = 11.11 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.11 * (11.11-4.24)(11.11-9.49)(11.11-8.49) } ; ; T = sqrt{ 324 } = 18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18 }{ 4.24 } = 8.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18 }{ 9.49 } = 3.79 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18 }{ 8.49 } = 4.24 ; ;

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{ 9.49**2+8.49**2-4.24**2 }{ 2 * 9.49 * 8.49 } ) = 26° 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+8.49**2-9.49**2 }{ 2 * 4.24 * 8.49 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 26° 33'54" - 90° = 63° 26'6" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18 }{ 11.11 } = 1.62 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4.24 }{ 2 * sin 26° 33'54" } = 4.74 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.49**2+2 * 8.49**2 - 4.24**2 } }{ 2 } = 8.746 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.49**2+2 * 4.24**2 - 9.49**2 } }{ 2 } = 4.743 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.49**2+2 * 4.24**2 - 8.49**2 } }{ 2 } = 6 ; ;
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