Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 8.48552813742   b = 16.97105627485   c = 18.9743665961

Area: T = 72
Perimeter: p = 44.43295100837
Semiperimeter: s = 22.21547550419

Angle ∠ A = α = 26.56550511771° = 26°33'54″ = 0.4643647609 rad
Angle ∠ B = β = 63.43549488229° = 63°26'6″ = 1.10771487178 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 16.97105627485
Height: hb = 8.48552813742
Height: hc = 7.58994663844

Median: ma = 17.49328556845
Median: mb = 12
Median: mc = 9.48768329805

Inradius: r = 3.24110890809
Circumradius: R = 9.48768329805

Vertex coordinates: A[9; 8] B[-9; 2] C[-3; -4]
Centroid: CG[-1; 2]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.62105445404; 3.24110890809]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.4354948823° = 153°26'6″ = 0.4643647609 rad
∠ B' = β' = 116.5655051177° = 116°33'54″ = 1.10771487178 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 = | beta gamma | = | beta - gamma | ; ; a**2 = ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 } ; ; a = sqrt{ (-9-(-3))**2 + (2-(-4))**2 } ; ; a = sqrt{ 72 } = 8.49 ; ;

2. We compute side b from coordinates using the Pythagorean theorem

b = | alpha gamma | = | alpha - gamma | ; ; b**2 = ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 } ; ; b = sqrt{ (9-(-3))**2 + (8-(-4))**2 } ; ; b = sqrt{ 288 } = 16.97 ; ;

3. We compute side c from coordinates using the Pythagorean theorem

c = | alpha beta | = | alpha - beta | ; ; c**2 = ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 } ; ; c = sqrt{ (9-(-9))**2 + (8-2)**2 } ; ; c = sqrt{ 360 } = 18.97 ; ;


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 = 16.97 ; ; c = 18.97 ; ;

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

p = a+b+c = 8.49+16.97+18.97 = 44.43 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 44.43 }{ 2 } = 22.21 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.21 * (22.21-8.49)(22.21-16.97)(22.21-18.97) } ; ; T = sqrt{ 5184 } = 72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 72 }{ 8.49 } = 16.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 72 }{ 16.97 } = 8.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 72 }{ 18.97 } = 7.59 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 8.49**2-16.97**2-18.97**2 }{ 2 * 16.97 * 18.97 } ) = 26° 33'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16.97**2-8.49**2-18.97**2 }{ 2 * 8.49 * 18.97 } ) = 63° 26'6" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 18.97**2-8.49**2-16.97**2 }{ 2 * 16.97 * 8.49 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 72 }{ 22.21 } = 3.24 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.49 }{ 2 * sin 26° 33'54" } = 9.49 ; ;




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