Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 4.12331056256   b = 18.02877563773   c = 14.21326704036

Area: T = 12.5
Perimeter: p = 36.36435324065
Semiperimeter: s = 18.18217662032

Angle ∠ A = α = 5.59993393365° = 5°35'58″ = 0.09877269074 rad
Angle ∠ B = β = 154.7476836605° = 154°44'49″ = 2.70108418058 rad
Angle ∠ C = γ = 19.65438240581° = 19°39'14″ = 0.34330239404 rad

Height: ha = 6.06333906259
Height: hb = 1.38767504906
Height: hc = 1.75989938618

Median: ma = 16.10112421881
Median: mb = 5.31550729064
Median: mc = 10.97772492001

Inradius: r = 0.68875019654
Circumradius: R = 21.12986535302

Vertex coordinates: A[8; 0] B[-3; 9] C[-7; 10]
Centroid: CG[-0.66766666667; 6.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-1.45875041667; 0.68875019654]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 174.4010660663° = 174°24'2″ = 0.09877269074 rad
∠ B' = β' = 25.25331633946° = 25°15'11″ = 2.70108418058 rad
∠ C' = γ' = 160.3466175942° = 160°20'46″ = 0.34330239404 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-(-7))**2 + (9-10)**2 } ; ; a = sqrt{ 17 } = 4.12 ; ;

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{ (8-(-7))**2 + (0-10)**2 } ; ; b = sqrt{ 325 } = 18.03 ; ;

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{ (8-(-3))**2 + (0-9)**2 } ; ; c = sqrt{ 202 } = 14.21 ; ;
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 = 4.12 ; ; b = 18.03 ; ; c = 14.21 ; ;

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

p = a+b+c = 4.12+18.03+14.21 = 36.36 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 36.36 }{ 2 } = 18.18 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.18 * (18.18-4.12)(18.18-18.03)(18.18-14.21) } ; ; T = sqrt{ 156.25 } = 12.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12.5 }{ 4.12 } = 6.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12.5 }{ 18.03 } = 1.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12.5 }{ 14.21 } = 1.76 ; ;

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{ 18.03**2+14.21**2-4.12**2 }{ 2 * 18.03 * 14.21 } ) = 5° 35'58" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.12**2+14.21**2-18.03**2 }{ 2 * 4.12 * 14.21 } ) = 154° 44'49" ; ;
gamma = 180° - alpha - beta = 180° - 5° 35'58" - 154° 44'49" = 19° 39'14" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12.5 }{ 18.18 } = 0.69 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4.12 }{ 2 * sin 5° 35'58" } = 21.13 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.03**2+2 * 14.21**2 - 4.12**2 } }{ 2 } = 16.101 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.21**2+2 * 4.12**2 - 18.03**2 } }{ 2 } = 5.315 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.03**2+2 * 4.12**2 - 14.21**2 } }{ 2 } = 10.977 ; ;
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