Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 10.19880390272   b = 8.24662112512   c = 13.4166407865

Area: T = 42
Perimeter: p = 31.86106581434
Semiperimeter: s = 15.93303290717

Angle ∠ A = α = 49.3998705355° = 49°23'55″ = 0.86221700547 rad
Angle ∠ B = β = 37.87549836511° = 37°52'30″ = 0.66110431689 rad
Angle ∠ C = γ = 92.72663109939° = 92°43'35″ = 1.61883794301 rad

Height: ha = 8.23768776758
Height: hb = 10.18664962515
Height: hc = 6.2610990337

Median: ma = 9.89994949366
Median: mb = 11.18803398875
Median: mc = 6.40331242374

Inradius: r = 2.63664803772
Circumradius: R = 6.71658052992

Vertex coordinates: A[-2; -4] B[4; 8] C[6; -2]
Centroid: CG[2.66766666667; 0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[3.3989760485; 2.63664803772]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.6011294645° = 130°36'5″ = 0.86221700547 rad
∠ B' = β' = 142.1255016349° = 142°7'30″ = 0.66110431689 rad
∠ C' = γ' = 87.27436890061° = 87°16'25″ = 1.61883794301 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{ (4-6)**2 + (8-(-2))**2 } ; ; a = sqrt{ 104 } = 10.2 ; ;

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 + (-4-(-2))**2 } ; ; b = sqrt{ 68 } = 8.25 ; ;

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{ (-2-4)**2 + (-4-8)**2 } ; ; c = sqrt{ 180 } = 13.42 ; ;


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 = 10.2 ; ; b = 8.25 ; ; c = 13.42 ; ;

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

p = a+b+c = 10.2+8.25+13.42 = 31.86 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.86 }{ 2 } = 15.93 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.93 * (15.93-10.2)(15.93-8.25)(15.93-13.42) } ; ; T = sqrt{ 1764 } = 42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 42 }{ 10.2 } = 8.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 42 }{ 8.25 } = 10.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 42 }{ 13.42 } = 6.26 ; ;

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{ 8.25**2+13.42**2-10.2**2 }{ 2 * 8.25 * 13.42 } ) = 49° 23'55" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.2**2+13.42**2-8.25**2 }{ 2 * 10.2 * 13.42 } ) = 37° 52'30" ; ; gamma = 180° - alpha - beta = 180° - 49° 23'55" - 37° 52'30" = 92° 43'35" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 42 }{ 15.93 } = 2.64 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10.2 }{ 2 * sin 49° 23'55" } = 6.72 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.25**2+2 * 13.42**2 - 10.2**2 } }{ 2 } = 9.899 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.42**2+2 * 10.2**2 - 8.25**2 } }{ 2 } = 11.18 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.25**2+2 * 10.2**2 - 13.42**2 } }{ 2 } = 6.403 ; ;
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