Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 19.64768827044   b = 11.04553610172   c = 20.88106130178

Area: T = 107
Perimeter: p = 51.57328567394
Semiperimeter: s = 25.78664283697

Angle ∠ A = α = 68.10663268583° = 68°6'23″ = 1.18986796451 rad
Angle ∠ B = β = 31.44328070705° = 31°26'34″ = 0.54987805094 rad
Angle ∠ C = γ = 80.45108660713° = 80°27'3″ = 1.4044132499 rad

Height: ha = 10.89223132092
Height: hb = 19.37546496531
Height: hc = 10.24987412519

Median: ma = 13.50992560861
Median: mb = 19.50664092031
Median: mc = 12.04215945788

Inradius: r = 4.14994695762
Circumradius: R = 10.58770031743

Vertex coordinates: A[-1; -3] B[5; 17] C[10; -2]
Centroid: CG[4.66766666667; 4]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[6.78765156621; 4.14994695762]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 111.8943673142° = 111°53'37″ = 1.18986796451 rad
∠ B' = β' = 148.557719293° = 148°33'26″ = 0.54987805094 rad
∠ C' = γ' = 99.54991339287° = 99°32'57″ = 1.4044132499 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{ (5-10)**2 + (17-(-2))**2 } ; ; a = sqrt{ 386 } = 19.65 ; ;

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{ (-1-10)**2 + (-3-(-2))**2 } ; ; b = sqrt{ 122 } = 11.05 ; ;

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{ (-1-5)**2 + (-3-17)**2 } ; ; c = sqrt{ 436 } = 20.88 ; ;
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 = 19.65 ; ; b = 11.05 ; ; c = 20.88 ; ;

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

p = a+b+c = 19.65+11.05+20.88 = 51.57 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51.57 }{ 2 } = 25.79 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.79 * (25.79-19.65)(25.79-11.05)(25.79-20.88) } ; ; T = sqrt{ 11449 } = 107 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 107 }{ 19.65 } = 10.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 107 }{ 11.05 } = 19.37 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 107 }{ 20.88 } = 10.25 ; ;

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{ 11.05**2+20.88**2-19.65**2 }{ 2 * 11.05 * 20.88 } ) = 68° 6'23" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 19.65**2+20.88**2-11.05**2 }{ 2 * 19.65 * 20.88 } ) = 31° 26'34" ; ; gamma = 180° - alpha - beta = 180° - 68° 6'23" - 31° 26'34" = 80° 27'3" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 107 }{ 25.79 } = 4.15 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 19.65 }{ 2 * sin 68° 6'23" } = 10.59 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.05**2+2 * 20.88**2 - 19.65**2 } }{ 2 } = 13.509 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.88**2+2 * 19.65**2 - 11.05**2 } }{ 2 } = 19.506 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.05**2+2 * 19.65**2 - 20.88**2 } }{ 2 } = 12.042 ; ;
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