Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 16.12545154966   b = 18.02877563773   c = 8.06222577483

Area: T = 65
Perimeter: p = 42.21545296222
Semiperimeter: s = 21.10772648111

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 = 8.06222577483
Height: hb = 7.21111025509
Height: hc = 16.12545154966

Median: ma = 11.4021754251
Median: mb = 9.01438781887
Median: mc = 16.62107701386

Inradius: r = 3.08795084338
Circumradius: R = 9.01438781887

Vertex coordinates: A[2; 6] B[-6; 7] C[-8; -9]
Centroid: CG[-4; 1.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 3.08795084338]

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{ (-6-(-8))**2 + (7-(-9))**2 } ; ; a = sqrt{ 260 } = 16.12 ; ;

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{ (2-(-8))**2 + (6-(-9))**2 } ; ; b = sqrt{ 325 } = 18.03 ; ;

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{ (2-(-6))**2 + (6-7)**2 } ; ; c = sqrt{ 65 } = 8.06 ; ;


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 = 16.12 ; ; b = 18.03 ; ; c = 8.06 ; ;

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

p = a+b+c = 16.12+18.03+8.06 = 42.21 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 42.21 }{ 2 } = 21.11 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.11 * (21.11-16.12)(21.11-18.03)(21.11-8.06) } ; ; T = sqrt{ 4225 } = 65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 65 }{ 16.12 } = 8.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 65 }{ 18.03 } = 7.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 65 }{ 8.06 } = 16.12 ; ;

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{ 16.12**2-18.03**2-8.06**2 }{ 2 * 18.03 * 8.06 } ) = 63° 26'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18.03**2-16.12**2-8.06**2 }{ 2 * 16.12 * 8.06 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.06**2-16.12**2-18.03**2 }{ 2 * 18.03 * 16.12 } ) = 26° 33'54" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 65 }{ 21.11 } = 3.08 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16.12 }{ 2 * sin 63° 26'6" } = 9.01 ; ;




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