Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 15.23215462117   b = 7.61657731059   c = 17.02993863659

Area: T = 58
Perimeter: p = 39.87767056835
Semiperimeter: s = 19.93883528418

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

Height: ha = 7.61657731059
Height: hb = 15.23215462117
Height: hc = 6.81217545464

Median: ma = 10.77703296143
Median: mb = 15.77003184681
Median: mc = 8.5154693183

Inradius: r = 2.90989664758
Circumradius: R = 8.5154693183

Vertex coordinates: A[8; -9] B[-5; 2] C[1; -12]
Centroid: CG[1.33333333333; -6.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[5.81879329517; 2.90989664758]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.5655051177° = 116°33'54″ = 1.10771487178 rad
∠ B' = β' = 153.4354948823° = 153°26'6″ = 0.4643647609 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{ (-5-1)**2 + (2-(-12))**2 } ; ; a = sqrt{ 232 } = 15.23 ; ;

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{ (8-1)**2 + (-9-(-12))**2 } ; ; b = sqrt{ 58 } = 7.62 ; ;

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{ (8-(-5))**2 + (-9-2)**2 } ; ; c = sqrt{ 290 } = 17.03 ; ;
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 = 15.23 ; ; b = 7.62 ; ; c = 17.03 ; ;

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

p = a+b+c = 15.23+7.62+17.03 = 39.88 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 39.88 }{ 2 } = 19.94 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.94 * (19.94-15.23)(19.94-7.62)(19.94-17.03) } ; ; T = sqrt{ 3364 } = 58 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 58 }{ 15.23 } = 7.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 58 }{ 7.62 } = 15.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 58 }{ 17.03 } = 6.81 ; ;

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{ 7.62**2+17.03**2-15.23**2 }{ 2 * 7.62 * 17.03 } ) = 63° 26'6" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 15.23**2+17.03**2-7.62**2 }{ 2 * 15.23 * 17.03 } ) = 26° 33'54" ; ; gamma = 180° - alpha - beta = 180° - 63° 26'6" - 26° 33'54" = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 58 }{ 19.94 } = 2.91 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 15.23 }{ 2 * sin 63° 26'6" } = 8.51 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.62**2+2 * 17.03**2 - 15.23**2 } }{ 2 } = 10.77 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.03**2+2 * 15.23**2 - 7.62**2 } }{ 2 } = 15.7 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.62**2+2 * 15.23**2 - 17.03**2 } }{ 2 } = 8.515 ; ;
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