Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 12.36993168769   b = 8.06222577483   c = 8.944427191

Area: T = 36
Perimeter: p = 29.37658465352
Semiperimeter: s = 14.68879232676

Angle ∠ A = α = 93.18798301199° = 93°10'47″ = 1.6266294832 rad
Angle ∠ B = β = 40.6011294645° = 40°36'5″ = 0.70986262721 rad
Angle ∠ C = γ = 46.21988752351° = 46°13'8″ = 0.80766715494 rad

Height: ha = 5.82108550009
Height: hb = 8.93105008904
Height: hc = 8.0549844719

Median: ma = 5.85223499554
Median: mb = 10.01224921973
Median: mc = 9.43439811321

Inradius: r = 2.45109931965
Circumradius: R = 6.19441953114

Vertex coordinates: A[-7; 2] B[-3; 10] C[0; -2]
Centroid: CG[-3.33333333333; 3.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.85994920626; 2.45109931965]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 86.82201698801° = 86°49'13″ = 1.6266294832 rad
∠ B' = β' = 139.3998705355° = 139°23'55″ = 0.70986262721 rad
∠ C' = γ' = 133.7811124765° = 133°46'52″ = 0.80766715494 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{ (-3-0)**2 + (10-(-2))**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{ (-7-0)**2 + (2-(-2))**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{ (-7-(-3))**2 + (2-10)**2 } ; ; c = sqrt{ 80 } = 8.94 ; ;


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 = 8.94 ; ;

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

p = a+b+c = 12.37+8.06+8.94 = 29.38 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29.38 }{ 2 } = 14.69 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.69 * (14.69-12.37)(14.69-8.06)(14.69-8.94) } ; ; T = sqrt{ 1296 } = 36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 36 }{ 12.37 } = 5.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 36 }{ 8.06 } = 8.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 36 }{ 8.94 } = 8.05 ; ;

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{ 12.37**2-8.06**2-8.94**2 }{ 2 * 8.06 * 8.94 } ) = 93° 10'47" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.06**2-12.37**2-8.94**2 }{ 2 * 12.37 * 8.94 } ) = 40° 36'5" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.94**2-12.37**2-8.06**2 }{ 2 * 8.06 * 12.37 } ) = 46° 13'8" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 36 }{ 14.69 } = 2.45 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.37 }{ 2 * sin 93° 10'47" } = 6.19 ; ;




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