Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 9.05553851381   b = 13.45436240471   c = 10.05498756211

Area: T = 45.5
Perimeter: p = 32.55988848063
Semiperimeter: s = 16.27994424032

Angle ∠ A = α = 42.30221943667° = 42°18'8″ = 0.73883125725 rad
Angle ∠ B = β = 89.37704013916° = 89°22'13″ = 1.56598077581 rad
Angle ∠ C = γ = 48.32774042417° = 48°19'39″ = 0.8433472323 rad

Height: ha = 10.04992688728
Height: hb = 6.76439767308
Height: hc = 9.05548384309

Median: ma = 10.97772492001
Median: mb = 6.80107352544
Median: mc = 10.3087764064

Inradius: r = 2.79549360226
Circumradius: R = 6.72772181707

Vertex coordinates: A[4; 3] B[-6; 2] C[-5; -7]
Centroid: CG[-2.33333333333; -0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.03107135827; 2.79549360226]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.6987805633° = 137°41'52″ = 0.73883125725 rad
∠ B' = β' = 90.63295986084° = 90°37'47″ = 1.56598077581 rad
∠ C' = γ' = 131.6732595758° = 131°40'21″ = 0.8433472323 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-(-5))**2 + (2-(-7))**2 } ; ; a = sqrt{ 82 } = 9.06 ; ;

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{ (4-(-5))**2 + (3-(-7))**2 } ; ; b = sqrt{ 181 } = 13.45 ; ;

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{ (4-(-6))**2 + (3-2)**2 } ; ; c = sqrt{ 101 } = 10.05 ; ;


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 = 9.06 ; ; b = 13.45 ; ; c = 10.05 ; ;

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

p = a+b+c = 9.06+13.45+10.05 = 32.56 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 32.56 }{ 2 } = 16.28 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.28 * (16.28-9.06)(16.28-13.45)(16.28-10.05) } ; ; T = sqrt{ 2070.25 } = 45.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 45.5 }{ 9.06 } = 10.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 45.5 }{ 13.45 } = 6.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 45.5 }{ 10.05 } = 9.05 ; ;

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{ 9.06**2-13.45**2-10.05**2 }{ 2 * 13.45 * 10.05 } ) = 42° 18'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13.45**2-9.06**2-10.05**2 }{ 2 * 9.06 * 10.05 } ) = 89° 22'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.05**2-9.06**2-13.45**2 }{ 2 * 13.45 * 9.06 } ) = 48° 19'39" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 45.5 }{ 16.28 } = 2.79 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.06 }{ 2 * sin 42° 18'8" } = 6.73 ; ;




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