Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 8.06222577483   b = 12.04215945788   c = 7.07110678119

Area: T = 27.5
Perimeter: p = 27.1754920139
Semiperimeter: s = 13.58774600695

Angle ∠ A = α = 40.23663583093° = 40°14'11″ = 0.70222569315 rad
Angle ∠ B = β = 105.2555118703° = 105°15'18″ = 1.83770483759 rad
Angle ∠ C = γ = 34.50985229877° = 34°30'31″ = 0.60222873461 rad

Height: ha = 6.82219104024
Height: hb = 4.5687501392
Height: hc = 7.77881745931

Median: ma = 9.01438781887
Median: mb = 4.61097722286
Median: mc = 9.61876920308

Inradius: r = 2.02439249911
Circumradius: R = 6.24106955535

Vertex coordinates: A[-3; 6] B[2; 1] C[9; 5]
Centroid: CG[2.66766666667; 4]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.5521979543; 2.02439249911]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.7643641691° = 139°45'49″ = 0.70222569315 rad
∠ B' = β' = 74.74548812969° = 74°44'42″ = 1.83770483759 rad
∠ C' = γ' = 145.4911477012° = 145°29'29″ = 0.60222873461 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{ (2-9)**2 + (1-5)**2 } ; ; a = sqrt{ 65 } = 8.06 ; ;

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{ (-3-9)**2 + (6-5)**2 } ; ; b = sqrt{ 145 } = 12.04 ; ;

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{ (-3-2)**2 + (6-1)**2 } ; ; c = sqrt{ 50 } = 7.07 ; ;


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 = 8.06 ; ; b = 12.04 ; ; c = 7.07 ; ;

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

p = a+b+c = 8.06+12.04+7.07 = 27.17 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.17 }{ 2 } = 13.59 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.59 * (13.59-8.06)(13.59-12.04)(13.59-7.07) } ; ; T = sqrt{ 756.25 } = 27.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 27.5 }{ 8.06 } = 6.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 27.5 }{ 12.04 } = 4.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 27.5 }{ 7.07 } = 7.78 ; ;

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{ 8.06**2-12.04**2-7.07**2 }{ 2 * 12.04 * 7.07 } ) = 40° 14'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.04**2-8.06**2-7.07**2 }{ 2 * 8.06 * 7.07 } ) = 105° 15'18" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.07**2-8.06**2-12.04**2 }{ 2 * 12.04 * 8.06 } ) = 34° 30'31" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 27.5 }{ 13.59 } = 2.02 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.06 }{ 2 * sin 40° 14'11" } = 6.24 ; ;




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