Triangle calculator VC

Please enter the coordinates of the three vertices


Right isosceles triangle.

Sides: a = 705.6932567624   b = 705.6932567624   c = 998

Area: T = 249001
Perimeter: p = 2409.385513525
Semiperimeter: s = 1204.693256762

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

Height: ha = 705.6932567624
Height: hb = 705.6932567624
Height: hc = 499

Median: ma = 788.9888276212
Median: mb = 788.9888276212
Median: mc = 499

Inradius: r = 206.6932567624
Circumradius: R = 499

Vertex coordinates: A[5000; 499] B[5000; -499] C[4501; 0]
Centroid: CG[4833.667666667; 0]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[206.6932567624; 206.6932567624]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 90° = 1.57107963268 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{ (5000-4501)**2 + (-499-0)**2 } ; ; a = sqrt{ 498002 } = 705.69 ; ;

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{ (5000-4501)**2 + (499-0)**2 } ; ; b = sqrt{ 498002 } = 705.69 ; ;

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{ (5000-5000)**2 + (499-(-499))**2 } ; ; c = sqrt{ 996004 } = 998 ; ;
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 = 705.69 ; ; b = 705.69 ; ; c = 998 ; ;

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

p = a+b+c = 705.69+705.69+998 = 2409.39 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2409.39 }{ 2 } = 1204.69 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1204.69 * (1204.69-705.69)(1204.69-705.69)(1204.69-998) } ; ; T = sqrt{ 62001498001 } = 249001 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 249001 }{ 705.69 } = 705.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 249001 }{ 705.69 } = 705.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 249001 }{ 998 } = 499 ; ;

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{ 705.69**2+998**2-705.69**2 }{ 2 * 705.69 * 998 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 705.69**2+998**2-705.69**2 }{ 2 * 705.69 * 998 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 45° - 45° = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 249001 }{ 1204.69 } = 206.69 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 705.69 }{ 2 * sin 45° } = 499 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 705.69**2+2 * 998**2 - 705.69**2 } }{ 2 } = 788.988 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 998**2+2 * 705.69**2 - 705.69**2 } }{ 2 } = 788.988 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 705.69**2+2 * 705.69**2 - 998**2 } }{ 2 } = 499 ; ;
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