Triangle calculator VC

Please enter the coordinates of the three vertices


Right isosceles triangle.

Sides: a = 9.22195444573   b = 13.03884048104   c = 9.22195444573

Area: T = 42.5
Perimeter: p = 31.4777493725
Semiperimeter: s = 15.73987468625

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 9.22195444573
Height: hb = 6.51992024052
Height: hc = 9.22195444573

Median: ma = 10.3087764064
Median: mb = 6.51992024052
Median: mc = 10.3087764064

Inradius: r = 2.77003420521
Circumradius: R = 6.51992024052

Vertex coordinates: A[-6; 11] B[3; 9] C[1; 0]
Centroid: CG[-0.66766666667; 6.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 2.77003420521]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 135° = 0.78553981634 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{ (3-1)**2 + (9-0)**2 } ; ; a = sqrt{ 85 } = 9.22 ; ;

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{ (-6-1)**2 + (11-0)**2 } ; ; b = sqrt{ 170 } = 13.04 ; ;

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{ (-6-3)**2 + (11-9)**2 } ; ; c = sqrt{ 85 } = 9.22 ; ;
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 = 9.22 ; ; b = 13.04 ; ; c = 9.22 ; ;

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

p = a+b+c = 9.22+13.04+9.22 = 31.48 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.48 }{ 2 } = 15.74 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.74 * (15.74-9.22)(15.74-13.04)(15.74-9.22) } ; ; T = sqrt{ 1806.25 } = 42.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 42.5 }{ 9.22 } = 9.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 42.5 }{ 13.04 } = 6.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 42.5 }{ 9.22 } = 9.22 ; ;

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{ 13.04**2+9.22**2-9.22**2 }{ 2 * 13.04 * 9.22 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9.22**2+9.22**2-13.04**2 }{ 2 * 9.22 * 9.22 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 45° - 90° = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 42.5 }{ 15.74 } = 2.7 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 9.22 }{ 2 * sin 45° } = 6.52 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.04**2+2 * 9.22**2 - 9.22**2 } }{ 2 } = 10.308 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.22**2+2 * 9.22**2 - 13.04**2 } }{ 2 } = 6.519 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.04**2+2 * 9.22**2 - 9.22**2 } }{ 2 } = 10.308 ; ;
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