Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 5.38551648071   b = 3.60655512755   c = 5.83109518948

Area: T = 9.5
Perimeter: p = 14.82216679774
Semiperimeter: s = 7.41108339887

Angle ∠ A = α = 64.65438240581° = 64°39'14″ = 1.12884221038 rad
Angle ∠ B = β = 37.23548339816° = 37°14'5″ = 0.65498704494 rad
Angle ∠ C = γ = 78.11113419604° = 78°6'41″ = 1.36333001004 rad

Height: ha = 3.52882114254
Height: hb = 5.27696518641
Height: hc = 3.25884731177

Median: ma = 4.03111288741
Median: mb = 5.31550729064
Median: mc = 3.53655339059

Inradius: r = 1.28219070046
Circumradius: R = 2.9799384383

Vertex coordinates: A[-1; 2] B[2; 7] C[-3; 5]
Centroid: CG[-0.66766666667; 4.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.68767197429; 1.28219070046]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115.3466175942° = 115°20'46″ = 1.12884221038 rad
∠ B' = β' = 142.7655166018° = 142°45'55″ = 0.65498704494 rad
∠ C' = γ' = 101.889865804° = 101°53'19″ = 1.36333001004 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-(-3))**2 + (7-5)**2 } ; ; a = sqrt{ 29 } = 5.39 ; ;

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 + (2-5)**2 } ; ; b = sqrt{ 13 } = 3.61 ; ;

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{ (-1-2)**2 + (2-7)**2 } ; ; c = sqrt{ 34 } = 5.83 ; ;


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 = 5.39 ; ; b = 3.61 ; ; c = 5.83 ; ;

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

p = a+b+c = 5.39+3.61+5.83 = 14.82 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 14.82 }{ 2 } = 7.41 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.41 * (7.41-5.39)(7.41-3.61)(7.41-5.83) } ; ; T = sqrt{ 90.25 } = 9.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9.5 }{ 5.39 } = 3.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9.5 }{ 3.61 } = 5.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9.5 }{ 5.83 } = 3.26 ; ;

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{ 5.39**2-3.61**2-5.83**2 }{ 2 * 3.61 * 5.83 } ) = 64° 39'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.61**2-5.39**2-5.83**2 }{ 2 * 5.39 * 5.83 } ) = 37° 14'5" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.83**2-5.39**2-3.61**2 }{ 2 * 3.61 * 5.39 } ) = 78° 6'41" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9.5 }{ 7.41 } = 1.28 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.39 }{ 2 * sin 64° 39'14" } = 2.98 ; ;




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