Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 13.03884048104   b = 19.10549731745   c = 6.70882039325

Area: T = 22.5
Perimeter: p = 38.85215819174
Semiperimeter: s = 19.42657909587

Angle ∠ A = α = 20.55660452196° = 20°33'22″ = 0.35987706703 rad
Angle ∠ B = β = 149.0366243468° = 149°2'10″ = 2.60111731533 rad
Angle ∠ C = γ = 10.40877113125° = 10°24'28″ = 0.182164883 rad

Height: ha = 3.45113424498
Height: hb = 2.35554076517
Height: hc = 6.70882039325

Median: ma = 12.7487548784
Median: mb = 4.03111288741
Median: mc = 16.00878105936

Inradius: r = 1.15882539958
Circumradius: R = 18.56766965888

Vertex coordinates: A[-9; -5] B[-6; 1] C[5; 8]
Centroid: CG[-3.33333333333; 1.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-1.93304233264; 1.15882539958]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.444395478° = 159°26'38″ = 0.35987706703 rad
∠ B' = β' = 30.96437565321° = 30°57'50″ = 2.60111731533 rad
∠ C' = γ' = 169.5922288688° = 169°35'32″ = 0.182164883 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{ (-6-5)**2 + (1-8)**2 } ; ; a = sqrt{ 170 } = 13.04 ; ;

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{ (-9-5)**2 + (-5-8)**2 } ; ; b = sqrt{ 365 } = 19.1 ; ;

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{ (-9-(-6))**2 + (-5-1)**2 } ; ; c = sqrt{ 45 } = 6.71 ; ;
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 = 13.04 ; ; b = 19.1 ; ; c = 6.71 ; ;

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

p = a+b+c = 13.04+19.1+6.71 = 38.85 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38.85 }{ 2 } = 19.43 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.43 * (19.43-13.04)(19.43-19.1)(19.43-6.71) } ; ; T = sqrt{ 506.25 } = 22.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22.5 }{ 13.04 } = 3.45 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22.5 }{ 19.1 } = 2.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22.5 }{ 6.71 } = 6.71 ; ;

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{ 19.1**2+6.71**2-13.04**2 }{ 2 * 19.1 * 6.71 } ) = 20° 33'22" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13.04**2+6.71**2-19.1**2 }{ 2 * 13.04 * 6.71 } ) = 149° 2'10" ; ;
 gamma = 180° - alpha - beta = 180° - 20° 33'22" - 149° 2'10" = 10° 24'28" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22.5 }{ 19.43 } = 1.16 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13.04 }{ 2 * sin 20° 33'22" } = 18.57 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.1**2+2 * 6.71**2 - 13.04**2 } }{ 2 } = 12.748 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.71**2+2 * 13.04**2 - 19.1**2 } }{ 2 } = 4.031 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.1**2+2 * 13.04**2 - 6.71**2 } }{ 2 } = 16.008 ; ;
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