Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 12.36993168769   b = 8.06222577483   c = 12.08330459736

Area: T = 46.5
Perimeter: p = 32.51546205987
Semiperimeter: s = 16.25773102994

Angle ∠ A = α = 72.68110615685° = 72°40'52″ = 1.26985238282 rad
Angle ∠ B = β = 38.48801982483° = 38°28'49″ = 0.67216061563 rad
Angle ∠ C = γ = 68.83987401832° = 68°50'19″ = 1.20114626691 rad

Height: ha = 7.51986043761
Height: hb = 11.53552303168
Height: hc = 7.6976734764

Median: ma = 8.20106097334
Median: mb = 11.54333963806
Median: mc = 8.5154693183

Inradius: r = 2.86602517356
Circumradius: R = 6.47883719258

Vertex coordinates: A[0; -2] B[-5; 9] C[-8; -3]
Centroid: CG[-4.33333333333; 1.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[3.59883812158; 2.86602517356]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.3198938432° = 107°19'8″ = 1.26985238282 rad
∠ B' = β' = 141.5219801752° = 141°31'11″ = 0.67216061563 rad
∠ C' = γ' = 111.1611259817° = 111°9'41″ = 1.20114626691 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{ (-5-(-8))**2 + (9-(-3))**2 } ; ; a = sqrt{ 153 } = 12.37 ; ;

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{ (0-(-8))**2 + (-2-(-3))**2 } ; ; b = sqrt{ 65 } = 8.06 ; ;

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{ (0-(-5))**2 + (-2-9)**2 } ; ; c = sqrt{ 146 } = 12.08 ; ;


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 = 12.37 ; ; b = 8.06 ; ; c = 12.08 ; ;

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

p = a+b+c = 12.37+8.06+12.08 = 32.51 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 32.51 }{ 2 } = 16.26 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.26 * (16.26-12.37)(16.26-8.06)(16.26-12.08) } ; ; T = sqrt{ 2162.25 } = 46.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 46.5 }{ 12.37 } = 7.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 46.5 }{ 8.06 } = 11.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 46.5 }{ 12.08 } = 7.7 ; ;

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{ 12.37**2-8.06**2-12.08**2 }{ 2 * 8.06 * 12.08 } ) = 72° 40'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.06**2-12.37**2-12.08**2 }{ 2 * 12.37 * 12.08 } ) = 38° 28'49" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.08**2-12.37**2-8.06**2 }{ 2 * 8.06 * 12.37 } ) = 68° 50'19" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 46.5 }{ 16.26 } = 2.86 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.37 }{ 2 * sin 72° 40'52" } = 6.48 ; ;




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