Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 7.81102496759   b = 9.05553851381   c = 8.06222577483

Area: T = 29.5
Perimeter: p = 24.92878925623
Semiperimeter: s = 12.46439462812

Angle ∠ A = α = 53.91549269571° = 53°54'54″ = 0.94109929914 rad
Angle ∠ B = β = 69.55504523892° = 69°33'2″ = 1.21438843904 rad
Angle ∠ C = γ = 56.53546206536° = 56°32'5″ = 0.98767152718 rad

Height: ha = 7.5544175916
Height: hb = 6.51554600384
Height: hc = 7.31880493408

Median: ma = 7.63221687612
Median: mb = 6.51992024052
Median: mc = 7.43330343737

Inradius: r = 2.36768266322
Circumradius: R = 4.8322217955

Vertex coordinates: A[8; -7] B[1; -3] C[7; 2]
Centroid: CG[5.33333333333; -2.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.88325455239; 2.36768266322]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.0855073043° = 126°5'6″ = 0.94109929914 rad
∠ B' = β' = 110.4549547611° = 110°26'58″ = 1.21438843904 rad
∠ C' = γ' = 123.4655379346° = 123°27'55″ = 0.98767152718 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{ (1-7)**2 + (-3-2)**2 } ; ; a = sqrt{ 61 } = 7.81 ; ;

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{ (8-7)**2 + (-7-2)**2 } ; ; b = sqrt{ 82 } = 9.06 ; ;

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{ (8-1)**2 + (-7-(-3))**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 = 7.81 ; ; b = 9.06 ; ; c = 8.06 ; ;

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

p = a+b+c = 7.81+9.06+8.06 = 24.93 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 24.93 }{ 2 } = 12.46 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.46 * (12.46-7.81)(12.46-9.06)(12.46-8.06) } ; ; T = sqrt{ 870.25 } = 29.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 29.5 }{ 7.81 } = 7.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 29.5 }{ 9.06 } = 6.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 29.5 }{ 8.06 } = 7.32 ; ;

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{ 7.81**2-9.06**2-8.06**2 }{ 2 * 9.06 * 8.06 } ) = 53° 54'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.06**2-7.81**2-8.06**2 }{ 2 * 7.81 * 8.06 } ) = 69° 33'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.06**2-7.81**2-9.06**2 }{ 2 * 9.06 * 7.81 } ) = 56° 32'5" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 29.5 }{ 12.46 } = 2.37 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.81 }{ 2 * sin 53° 54'54" } = 4.83 ; ;




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