Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 7.28801098893   b = 16.27988205961   c = 14.56602197786

Area: T = 53
Perimeter: p = 38.11991502639
Semiperimeter: s = 19.0659575132

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 = 14.56602197786
Height: hb = 6.51215282384
Height: hc = 7.28801098893

Median: ma = 15.00883310198
Median: mb = 8.1399410298
Median: mc = 10.2965630141

Inradius: r = 2.78107545359
Circumradius: R = 8.1399410298

Vertex coordinates: A[5; 1] B[-9; 5] C[-7; 12]
Centroid: CG[-3.66766666667; 6]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 2.78107545359]

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{ (-9-(-7))**2 + (5-12)**2 } ; ; a = sqrt{ 53 } = 7.28 ; ;

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-(-7))**2 + (1-12)**2 } ; ; b = sqrt{ 265 } = 16.28 ; ;

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-(-9))**2 + (1-5)**2 } ; ; c = sqrt{ 212 } = 14.56 ; ;


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 = 7.28 ; ; b = 16.28 ; ; c = 14.56 ; ;

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

p = a+b+c = 7.28+16.28+14.56 = 38.12 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38.12 }{ 2 } = 19.06 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.06 * (19.06-7.28)(19.06-16.28)(19.06-14.56) } ; ; T = sqrt{ 2809 } = 53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 53 }{ 7.28 } = 14.56 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 53 }{ 16.28 } = 6.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 53 }{ 14.56 } = 7.28 ; ;

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{ 16.28**2+14.56**2-7.28**2 }{ 2 * 16.28 * 14.56 } ) = 26° 33'54" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.28**2+14.56**2-16.28**2 }{ 2 * 7.28 * 14.56 } ) = 90° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 7.28**2+16.28**2-14.56**2 }{ 2 * 7.28 * 16.28 } ) = 63° 26'6" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 53 }{ 19.06 } = 2.78 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.28 }{ 2 * sin 26° 33'54" } = 8.14 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.28**2+2 * 14.56**2 - 7.28**2 } }{ 2 } = 15.008 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.56**2+2 * 7.28**2 - 16.28**2 } }{ 2 } = 8.139 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.28**2+2 * 7.28**2 - 14.56**2 } }{ 2 } = 10.296 ; ;
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