Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 12.80662484749   b = 14.31878210633   c = 6.40331242374

Area: T = 41
Perimeter: p = 33.52771937756
Semiperimeter: s = 16.76435968878

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

Height: ha = 6.40331242374
Height: hb = 5.72771284253
Height: hc = 12.80662484749

Median: ma = 9.05553851381
Median: mb = 7.15989105316
Median: mc = 13.22003787824

Inradius: r = 2.44657758245
Circumradius: R = 7.15989105316

Vertex coordinates: A[5; 1] B[0; 5] C[-8; -5]
Centroid: CG[-1; 0.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 2.44657758245]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.5655051177° = 116°33'54″ = 1.10771487178 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 153.4354948823° = 153°26'6″ = 0.4643647609 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{ (0-(-8))**2 + (5-(-5))**2 } ; ; a = sqrt{ 164 } = 12.81 ; ;

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{ (5-(-8))**2 + (1-(-5))**2 } ; ; b = sqrt{ 205 } = 14.32 ; ;

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{ (5-0)**2 + (1-5)**2 } ; ; c = sqrt{ 41 } = 6.4 ; ;


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 = 12.81 ; ; b = 14.32 ; ; c = 6.4 ; ;

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

p = a+b+c = 12.81+14.32+6.4 = 33.53 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33.53 }{ 2 } = 16.76 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.76 * (16.76-12.81)(16.76-14.32)(16.76-6.4) } ; ; T = sqrt{ 1681 } = 41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 41 }{ 12.81 } = 6.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 41 }{ 14.32 } = 5.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 41 }{ 6.4 } = 12.81 ; ;

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{ 14.32**2+6.4**2-12.81**2 }{ 2 * 14.32 * 6.4 } ) = 63° 26'6" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.81**2+6.4**2-14.32**2 }{ 2 * 12.81 * 6.4 } ) = 90° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 12.81**2+14.32**2-6.4**2 }{ 2 * 12.81 * 14.32 } ) = 26° 33'54" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 41 }{ 16.76 } = 2.45 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.81 }{ 2 * sin 63° 26'6" } = 7.16 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.32**2+2 * 6.4**2 - 12.81**2 } }{ 2 } = 9.055 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.4**2+2 * 12.81**2 - 14.32**2 } }{ 2 } = 7.159 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.32**2+2 * 12.81**2 - 6.4**2 } }{ 2 } = 13.2 ; ;
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