Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 6.40331242374   b = 45.27769256907   c = 44.8221869662

Area: T = 143.5
Perimeter: p = 96.50219195901
Semiperimeter: s = 48.25109597951

Angle ∠ A = α = 8.13301023542° = 8°7'48″ = 0.14218970546 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 81.87698976458° = 81°52'12″ = 1.42988992722 rad

Height: ha = 44.8221869662
Height: hb = 6.33987695967
Height: hc = 6.40331242374

Median: ma = 44.93660656934
Median: mb = 22.63884628453
Median: mc = 23.30877240416

Inradius: r = 2.97440341044
Circumradius: R = 22.63884628453

Vertex coordinates: A[12; -14] B[-16; 21] C[-11; 25]
Centroid: CG[-5; 10.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 2.97440341044]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 171.8769897646° = 171°52'12″ = 0.14218970546 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 98.13301023542° = 98°7'48″ = 1.42988992722 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{ (-16-(-11))**2 + (21-25)**2 } ; ; a = sqrt{ 41 } = 6.4 ; ;

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{ (12-(-11))**2 + (-14-25)**2 } ; ; b = sqrt{ 2050 } = 45.28 ; ;

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{ (12-(-16))**2 + (-14-21)**2 } ; ; c = sqrt{ 2009 } = 44.82 ; ;
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 = 6.4 ; ; b = 45.28 ; ; c = 44.82 ; ;

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

p = a+b+c = 6.4+45.28+44.82 = 96.5 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 96.5 }{ 2 } = 48.25 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 48.25 * (48.25-6.4)(48.25-45.28)(48.25-44.82) } ; ; T = sqrt{ 20592.25 } = 143.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 143.5 }{ 6.4 } = 44.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 143.5 }{ 45.28 } = 6.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 143.5 }{ 44.82 } = 6.4 ; ;

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{ 45.28**2+44.82**2-6.4**2 }{ 2 * 45.28 * 44.82 } ) = 8° 7'48" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.4**2+44.82**2-45.28**2 }{ 2 * 6.4 * 44.82 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 8° 7'48" - 90° = 81° 52'12" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 143.5 }{ 48.25 } = 2.97 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.4 }{ 2 * sin 8° 7'48" } = 22.64 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 45.28**2+2 * 44.82**2 - 6.4**2 } }{ 2 } = 44.936 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 44.82**2+2 * 6.4**2 - 45.28**2 } }{ 2 } = 22.638 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 45.28**2+2 * 6.4**2 - 44.82**2 } }{ 2 } = 23.308 ; ;
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