Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 7.28801098893   b = 7.61657731059   c = 6.40331242374

Area: T = 21.5
Perimeter: p = 21.29990072326
Semiperimeter: s = 10.65495036163

Angle ∠ A = α = 61.85883987677° = 61°51'30″ = 1.08796327285 rad
Angle ∠ B = β = 67.28655876468° = 67°17'8″ = 1.17443550436 rad
Angle ∠ C = γ = 50.85660135854° = 50°51'22″ = 0.88876048815 rad

Height: ha = 5.90765042498
Height: hb = 5.6466176613
Height: hc = 6.71554717612

Median: ma = 6.02107972894
Median: mb = 5.70108771255
Median: mc = 6.72768120235

Inradius: r = 2.019887344
Circumradius: R = 4.12880543701

Vertex coordinates: A[-4; -1] B[1; 3] C[3; -4]
Centroid: CG[0; -0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.84551098121; 2.019887344]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 118.1421601232° = 118°8'30″ = 1.08796327285 rad
∠ B' = β' = 112.7144412353° = 112°42'52″ = 1.17443550436 rad
∠ C' = γ' = 129.1443986415° = 129°8'38″ = 0.88876048815 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{ (1-3)**2 + (3-(-4))**2 } ; ; a = sqrt{ 53 } = 7.28 ; ;

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{ (-4-3)**2 + (-1-(-4))**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{ (-4-1)**2 + (-1-3)**2 } ; ; c = sqrt{ 41 } = 6.4 ; ;
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 = 7.28 ; ; b = 7.62 ; ; c = 6.4 ; ;

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

p = a+b+c = 7.28+7.62+6.4 = 21.3 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.3 }{ 2 } = 10.65 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.65 * (10.65-7.28)(10.65-7.62)(10.65-6.4) } ; ; T = sqrt{ 462.25 } = 21.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21.5 }{ 7.28 } = 5.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21.5 }{ 7.62 } = 5.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21.5 }{ 6.4 } = 6.72 ; ;

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+6.4**2-7.28**2 }{ 2 * 7.62 * 6.4 } ) = 61° 51'30" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.28**2+6.4**2-7.62**2 }{ 2 * 7.28 * 6.4 } ) = 67° 17'8" ; ;
 gamma = 180° - alpha - beta = 180° - 61° 51'30" - 67° 17'8" = 50° 51'22" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21.5 }{ 10.65 } = 2.02 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.28 }{ 2 * sin 61° 51'30" } = 4.13 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.62**2+2 * 6.4**2 - 7.28**2 } }{ 2 } = 6.021 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.4**2+2 * 7.28**2 - 7.62**2 } }{ 2 } = 5.701 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.62**2+2 * 7.28**2 - 6.4**2 } }{ 2 } = 6.727 ; ;
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