Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 1.41442135624   b = 8.6022325267   c = 8.48552813742

Area: T = 6
Perimeter: p = 18.50218202037
Semiperimeter: s = 9.25109101018

Angle ∠ A = α = 9.4622322208° = 9°27'44″ = 0.16551486774 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 80.5387677792° = 80°32'16″ = 1.40656476494 rad

Height: ha = 8.48552813742
Height: hb = 1.39549716649
Height: hc = 1.41442135624

Median: ma = 8.5154693183
Median: mb = 4.30111626335
Median: mc = 4.4722135955

Inradius: r = 0.64985848348
Circumradius: R = 4.30111626335

Vertex coordinates: A[-3; -1] B[3; 5] C[2; 6]
Centroid: CG[0.66766666667; 3.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 0.64985848348]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170.5387677792° = 170°32'16″ = 0.16551486774 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 99.4622322208° = 99°27'44″ = 1.40656476494 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{ (3-2)**2 + (5-6)**2 } ; ; a = sqrt{ 2 } = 1.41 ; ;

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{ (-3-2)**2 + (-1-6)**2 } ; ; b = sqrt{ 74 } = 8.6 ; ;

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{ (-3-3)**2 + (-1-5)**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 = 1.41 ; ; b = 8.6 ; ; c = 8.49 ; ;

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

p = a+b+c = 1.41+8.6+8.49 = 18.5 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.5 }{ 2 } = 9.25 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.25 * (9.25-1.41)(9.25-8.6)(9.25-8.49) } ; ; T = sqrt{ 36 } = 6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6 }{ 1.41 } = 8.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6 }{ 8.6 } = 1.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6 }{ 8.49 } = 1.41 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 8.6**2+8.49**2-1.41**2 }{ 2 * 8.6 * 8.49 } ) = 9° 27'44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.41**2+8.49**2-8.6**2 }{ 2 * 1.41 * 8.49 } ) = 90° ; ;
 gamma = 180° - alpha - beta = 180° - 9° 27'44" - 90° = 80° 32'16" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6 }{ 9.25 } = 0.65 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.41 }{ 2 * sin 9° 27'44" } = 4.3 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.6**2+2 * 8.49**2 - 1.41**2 } }{ 2 } = 8.515 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.49**2+2 * 1.41**2 - 8.6**2 } }{ 2 } = 4.301 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.6**2+2 * 1.41**2 - 8.49**2 } }{ 2 } = 4.472 ; ;
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