Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 5.09990195136   b = 11.4021754251   c = 10.19880390272

Area: T = 26
Perimeter: p = 26.69988127918
Semiperimeter: s = 13.34994063959

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

Height: ha = 10.19880390272
Height: hb = 4.56107017004
Height: hc = 5.09990195136

Median: ma = 10.51218980208
Median: mb = 5.70108771255
Median: mc = 7.21111025509

Inradius: r = 1.94876521449
Circumradius: R = 5.70108771255

Vertex coordinates: A[-6; 0] B[4; -2] C[5; 3]
Centroid: CG[1; 0.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 1.94876521449]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.4354948823° = 153°26'6″ = 0.4643647609 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 116.5655051177° = 116°33'54″ = 1.10771487178 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{ (4-5)**2 + (-2-3)**2 } ; ; a = sqrt{ 26 } = 5.1 ; ;

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{ (-6-5)**2 + (0-3)**2 } ; ; b = sqrt{ 130 } = 11.4 ; ;

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{ (-6-4)**2 + (0-(-2))**2 } ; ; c = sqrt{ 104 } = 10.2 ; ;


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 = 5.1 ; ; b = 11.4 ; ; c = 10.2 ; ;

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

p = a+b+c = 5.1+11.4+10.2 = 26.7 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.7 }{ 2 } = 13.35 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.35 * (13.35-5.1)(13.35-11.4)(13.35-10.2) } ; ; T = sqrt{ 676 } = 26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 26 }{ 5.1 } = 10.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 26 }{ 11.4 } = 4.56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 26 }{ 10.2 } = 5.1 ; ;

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{ 5.1**2-11.4**2-10.2**2 }{ 2 * 11.4 * 10.2 } ) = 26° 33'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11.4**2-5.1**2-10.2**2 }{ 2 * 5.1 * 10.2 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.2**2-5.1**2-11.4**2 }{ 2 * 11.4 * 5.1 } ) = 63° 26'6" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 26 }{ 13.35 } = 1.95 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.1 }{ 2 * sin 26° 33'54" } = 5.7 ; ;




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