Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 8.06222577483   b = 6.70882039325   c = 4.4722135955

Area: T = 15
Perimeter: p = 19.24325976358
Semiperimeter: s = 9.62112988179

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 56.3109932474° = 56°18'36″ = 0.98327937232 rad
Angle ∠ C = γ = 33.6990067526° = 33°41'24″ = 0.58880026035 rad

Height: ha = 3.72110420377
Height: hb = 4.4722135955
Height: hc = 6.70882039325

Median: ma = 4.03111288741
Median: mb = 5.59901699437
Median: mc = 7.07110678119

Inradius: r = 1.55990410696
Circumradius: R = 4.03111288741

Vertex coordinates: A[8; -4] B[4; -6] C[11; -10]
Centroid: CG[7.66766666667; -6.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.03993607131; 1.55990410696]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 123.6990067526° = 123°41'24″ = 0.98327937232 rad
∠ C' = γ' = 146.3109932474° = 146°18'36″ = 0.58880026035 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{ (4-11)**2 + (-6-(-10))**2 } ; ; a = sqrt{ 65 } = 8.06 ; ;

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{ (8-11)**2 + (-4-(-10))**2 } ; ; b = sqrt{ 45 } = 6.71 ; ;

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{ (8-4)**2 + (-4-(-6))**2 } ; ; c = sqrt{ 20 } = 4.47 ; ;


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 = 8.06 ; ; b = 6.71 ; ; c = 4.47 ; ;

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

p = a+b+c = 8.06+6.71+4.47 = 19.24 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 19.24 }{ 2 } = 9.62 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.62 * (9.62-8.06)(9.62-6.71)(9.62-4.47) } ; ; T = sqrt{ 225 } = 15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 15 }{ 8.06 } = 3.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 15 }{ 6.71 } = 4.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 15 }{ 4.47 } = 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{ 6.71**2+4.47**2-8.06**2 }{ 2 * 6.71 * 4.47 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.06**2+4.47**2-6.71**2 }{ 2 * 8.06 * 4.47 } ) = 56° 18'36" ; ; gamma = 180° - alpha - beta = 180° - 90° - 56° 18'36" = 33° 41'24" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 15 }{ 9.62 } = 1.56 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.06 }{ 2 * sin 90° } = 4.03 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.71**2+2 * 4.47**2 - 8.06**2 } }{ 2 } = 4.031 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.47**2+2 * 8.06**2 - 6.71**2 } }{ 2 } = 5.59 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.71**2+2 * 8.06**2 - 4.47**2 } }{ 2 } = 7.071 ; ;
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