Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 3.16222776602   b = 1.41442135624   c = 2.82884271247

Area: T = 2
Perimeter: p = 7.40549183473
Semiperimeter: s = 3.70224591736

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 26.56550511771° = 26°33'54″ = 0.4643647609 rad
Angle ∠ C = γ = 63.43549488229° = 63°26'6″ = 1.10771487178 rad

Height: ha = 1.26549110641
Height: hb = 2.82884271247
Height: hc = 1.41442135624

Median: ma = 1.58111388301
Median: mb = 2.91554759474
Median: mc = 2

Inradius: r = 0.54401815135
Circumradius: R = 1.58111388301

Vertex coordinates: A[6; 4] B[4; 6] C[7; 5]
Centroid: CG[5.66766666667; 5]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.0880363027; 0.54401815135]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 153.4354948823° = 153°26'6″ = 0.4643647609 rad
∠ C' = γ' = 116.5655051177° = 116°33'54″ = 1.10771487178 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{ (4-7)**2 + (6-5)**2 } ; ; a = sqrt{ 10 } = 3.16 ; ;

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{ (6-7)**2 + (4-5)**2 } ; ; b = sqrt{ 2 } = 1.41 ; ;

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{ (6-4)**2 + (4-6)**2 } ; ; c = sqrt{ 8 } = 2.83 ; ;
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 = 3.16 ; ; b = 1.41 ; ; c = 2.83 ; ;

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

p = a+b+c = 3.16+1.41+2.83 = 7.4 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 7.4 }{ 2 } = 3.7 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.7 * (3.7-3.16)(3.7-1.41)(3.7-2.83) } ; ; T = sqrt{ 4 } = 2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2 }{ 3.16 } = 1.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2 }{ 1.41 } = 2.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2 }{ 2.83 } = 1.41 ; ;

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{ 1.41**2+2.83**2-3.16**2 }{ 2 * 1.41 * 2.83 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.16**2+2.83**2-1.41**2 }{ 2 * 3.16 * 2.83 } ) = 26° 33'54" ; ;
 gamma = 180° - alpha - beta = 180° - 90° - 26° 33'54" = 63° 26'6" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2 }{ 3.7 } = 0.54 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.16 }{ 2 * sin 90° } = 1.58 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.41**2+2 * 2.83**2 - 3.16**2 } }{ 2 } = 1.581 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.83**2+2 * 3.16**2 - 1.41**2 } }{ 2 } = 2.915 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.41**2+2 * 3.16**2 - 2.83**2 } }{ 2 } = 2 ; ;
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