Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 6.32545553203   b = 6.40331242374   c = 7.28801098893

Area: T = 19
Perimeter: p = 20.00877894471
Semiperimeter: s = 10.00438947235

Angle ∠ A = α = 54.6055204155° = 54°36'19″ = 0.95330406012 rad
Angle ∠ B = β = 55.62196552762° = 55°37'11″ = 0.97107461134 rad
Angle ∠ C = γ = 69.77551405688° = 69°46'31″ = 1.2187805939 rad

Height: ha = 6.00883275543
Height: hb = 5.93546029518
Height: hc = 5.22197014301

Median: ma = 6.08327625303
Median: mb = 6.02107972894
Median: mc = 5.22201532545

Inradius: r = 1.89992602906
Circumradius: R = 3.87992365814

Vertex coordinates: A[1; 5] B[3; -2] C[-3; 0]
Centroid: CG[0.33333333333; 1]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.29994938831; 1.89992602906]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.3954795845° = 125°23'41″ = 0.95330406012 rad
∠ B' = β' = 124.3880344724° = 124°22'49″ = 0.97107461134 rad
∠ C' = γ' = 110.2254859431° = 110°13'29″ = 1.2187805939 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{ (3-(-3))**2 + (-2-0)**2 } ; ; a = sqrt{ 40 } = 6.32 ; ;

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{ (1-(-3))**2 + (5-0)**2 } ; ; b = sqrt{ 41 } = 6.4 ; ;

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{ (1-3)**2 + (5-(-2))**2 } ; ; c = sqrt{ 53 } = 7.28 ; ;


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 = 6.32 ; ; b = 6.4 ; ; c = 7.28 ; ;

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

p = a+b+c = 6.32+6.4+7.28 = 20.01 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.01 }{ 2 } = 10 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10 * (10-6.32)(10-6.4)(10-7.28) } ; ; T = sqrt{ 361 } = 19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 19 }{ 6.32 } = 6.01 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 19 }{ 6.4 } = 5.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 19 }{ 7.28 } = 5.22 ; ;

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{ 6.32**2-6.4**2-7.28**2 }{ 2 * 6.4 * 7.28 } ) = 54° 36'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.4**2-6.32**2-7.28**2 }{ 2 * 6.32 * 7.28 } ) = 55° 37'11" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.28**2-6.32**2-6.4**2 }{ 2 * 6.4 * 6.32 } ) = 69° 46'31" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 19 }{ 10 } = 1.9 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.32 }{ 2 * sin 54° 36'19" } = 3.88 ; ;




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