Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 7.61657731059   b = 8.54440037453   c = 6.08327625303

Area: T = 22.5
Perimeter: p = 22.24325393815
Semiperimeter: s = 11.12112696907

Angle ∠ A = α = 59.98216325724° = 59°58'54″ = 1.04768769791 rad
Angle ∠ B = β = 76.26437316944° = 76°15'49″ = 1.33110532179 rad
Angle ∠ C = γ = 43.75546357332° = 43°45'17″ = 0.76436624566 rad

Height: ha = 5.90987894787
Height: hb = 5.26768516238
Height: hc = 7.39879544287

Median: ma = 6.36439610307
Median: mb = 5.40883269132
Median: mc = 7.5

Inradius: r = 2.023315029
Circumradius: R = 4.39877828308

Vertex coordinates: A[0; 0] B[6; 1] C[3; 8]
Centroid: CG[3; 3]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.49545478487; 2.023315029]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120.0188367428° = 120°1'6″ = 1.04768769791 rad
∠ B' = β' = 103.7366268306° = 103°44'11″ = 1.33110532179 rad
∠ C' = γ' = 136.2455364267° = 136°14'43″ = 0.76436624566 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{ (6-3)**2 + (1-8)**2 } ; ; a = sqrt{ 58 } = 7.62 ; ;

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{ (0-3)**2 + (0-8)**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{ (0-6)**2 + (0-1)**2 } ; ; c = sqrt{ 37 } = 6.08 ; ;
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 = 7.62 ; ; b = 8.54 ; ; c = 6.08 ; ;

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

p = a+b+c = 7.62+8.54+6.08 = 22.24 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.24 }{ 2 } = 11.12 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.12 * (11.12-7.62)(11.12-8.54)(11.12-6.08) } ; ; 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 }{ 7.62 } = 5.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22.5 }{ 8.54 } = 5.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22.5 }{ 6.08 } = 7.4 ; ;

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+6.08**2-7.62**2 }{ 2 * 8.54 * 6.08 } ) = 59° 58'54" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.62**2+6.08**2-8.54**2 }{ 2 * 7.62 * 6.08 } ) = 76° 15'49" ; ;
 gamma = 180° - alpha - beta = 180° - 59° 58'54" - 76° 15'49" = 43° 45'17" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22.5 }{ 11.12 } = 2.02 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.62 }{ 2 * sin 59° 58'54" } = 4.4 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.54**2+2 * 6.08**2 - 7.62**2 } }{ 2 } = 6.364 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.08**2+2 * 7.62**2 - 8.54**2 } }{ 2 } = 5.408 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.54**2+2 * 7.62**2 - 6.08**2 } }{ 2 } = 7.5 ; ;
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