Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 7.81102496759   b = 8.54440037453   c = 4.24326406871

Area: T = 16.5
Perimeter: p = 20.59768941083
Semiperimeter: s = 10.29884470542

Angle ∠ A = α = 65.55660452196° = 65°33'22″ = 1.14441688337 rad
Angle ∠ B = β = 84.80655710923° = 84°48'20″ = 1.48801364396 rad
Angle ∠ C = γ = 29.63883836882° = 29°38'18″ = 0.51772873803 rad

Height: ha = 4.22552170378
Height: hb = 3.86223578575
Height: hc = 7.77881745931

Median: ma = 5.5
Median: mb = 4.61097722286
Median: mc = 7.90656941504

Inradius: r = 1.60221833111
Circumradius: R = 4.298961845

Vertex coordinates: A[-2; 2] B[1; 5] C[6; -1]
Centroid: CG[1.66766666667; 2]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.14656530283; 1.60221833111]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 114.444395478° = 114°26'38″ = 1.14441688337 rad
∠ B' = β' = 95.19444289077° = 95°11'40″ = 1.48801364396 rad
∠ C' = γ' = 150.3621616312° = 150°21'42″ = 0.51772873803 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{ (1-6)**2 + (5-(-1))**2 } ; ; a = sqrt{ 61 } = 7.81 ; ;

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{ (-2-6)**2 + (2-(-1))**2 } ; ; b = sqrt{ 73 } = 8.54 ; ;

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{ (-2-1)**2 + (2-5)**2 } ; ; c = sqrt{ 18 } = 4.24 ; ;
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 = 7.81 ; ; b = 8.54 ; ; c = 4.24 ; ;

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

p = a+b+c = 7.81+8.54+4.24 = 20.6 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.6 }{ 2 } = 10.3 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.3 * (10.3-7.81)(10.3-8.54)(10.3-4.24) } ; ; T = sqrt{ 272.25 } = 16.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16.5 }{ 7.81 } = 4.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16.5 }{ 8.54 } = 3.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16.5 }{ 4.24 } = 7.78 ; ;

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{ 8.54**2+4.24**2-7.81**2 }{ 2 * 8.54 * 4.24 } ) = 65° 33'22" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.81**2+4.24**2-8.54**2 }{ 2 * 7.81 * 4.24 } ) = 84° 48'20" ; ; gamma = 180° - alpha - beta = 180° - 65° 33'22" - 84° 48'20" = 29° 38'18" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 16.5 }{ 10.3 } = 1.6 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.81 }{ 2 * sin 65° 33'22" } = 4.29 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.54**2+2 * 4.24**2 - 7.81**2 } }{ 2 } = 5.5 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.24**2+2 * 7.81**2 - 8.54**2 } }{ 2 } = 4.61 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.54**2+2 * 7.81**2 - 4.24**2 } }{ 2 } = 7.906 ; ;
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