Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 8.48552813742   b = 10.19880390272   c = 5.65768542495

Area: T = 24
Perimeter: p = 24.34401746509
Semiperimeter: s = 12.17700873255

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

Height: ha = 5.65768542495
Height: hb = 4.70767872433
Height: hc = 8.48552813742

Median: ma = 7.07110678119
Median: mb = 5.09990195136
Median: mc = 8.944427191

Inradius: r = 1.97220482983
Circumradius: R = 5.09990195136

Vertex coordinates: A[0; 0] B[-4; -4] C[-10; 2]
Centroid: CG[-4.66766666667; -0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 1.97220482983]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.6990067526° = 123°41'24″ = 0.98327937232 rad
∠ B' = β' = 90° = 1.57107963268 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-(-10))**2 + (-4-2)**2 } ; ; a = sqrt{ 72 } = 8.49 ; ;

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-(-10))**2 + (0-2)**2 } ; ; b = sqrt{ 104 } = 10.2 ; ;

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-(-4))**2 + (0-(-4))**2 } ; ; c = sqrt{ 32 } = 5.66 ; ;
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 = 8.49 ; ; b = 10.2 ; ; c = 5.66 ; ;

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

p = a+b+c = 8.49+10.2+5.66 = 24.34 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 24.34 }{ 2 } = 12.17 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.17 * (12.17-8.49)(12.17-10.2)(12.17-5.66) } ; ; T = sqrt{ 576 } = 24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 24 }{ 8.49 } = 5.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 24 }{ 10.2 } = 4.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 24 }{ 5.66 } = 8.49 ; ;

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{ 10.2**2+5.66**2-8.49**2 }{ 2 * 10.2 * 5.66 } ) = 56° 18'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.49**2+5.66**2-10.2**2 }{ 2 * 8.49 * 5.66 } ) = 90° ; ;
 gamma = 180° - alpha - beta = 180° - 56° 18'36" - 90° = 33° 41'24" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 24 }{ 12.17 } = 1.97 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.49 }{ 2 * sin 56° 18'36" } = 5.1 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.2**2+2 * 5.66**2 - 8.49**2 } }{ 2 } = 7.071 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.66**2+2 * 8.49**2 - 10.2**2 } }{ 2 } = 5.099 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.2**2+2 * 8.49**2 - 5.66**2 } }{ 2 } = 8.944 ; ;
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