Triangle calculator VC

Please enter the coordinates of the three vertices


Acute isosceles triangle.

Sides: a = 5.65768542495   b = 5.09990195136   c = 5.09990195136

Area: T = 12
Perimeter: p = 15.85548932767
Semiperimeter: s = 7.92774466383

Angle ∠ A = α = 67.3880135052° = 67°22'49″ = 1.17660052071 rad
Angle ∠ B = β = 56.3109932474° = 56°18'36″ = 0.98327937232 rad
Angle ∠ C = γ = 56.3109932474° = 56°18'36″ = 0.98327937232 rad

Height: ha = 4.24326406871
Height: hb = 4.70767872433
Height: hc = 4.70767872433

Median: ma = 4.24326406871
Median: mb = 4.74334164903
Median: mc = 4.74334164903

Inradius: r = 1.51437282592
Circumradius: R = 3.06441293851

Vertex coordinates: A[-3; 2] B[2; 1] C[-2; -3]
Centroid: CG[-1; 0]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.00991521728; 1.51437282592]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.6219864948° = 112°37'11″ = 1.17660052071 rad
∠ B' = β' = 123.6990067526° = 123°41'24″ = 0.98327937232 rad
∠ C' = γ' = 123.6990067526° = 123°41'24″ = 0.98327937232 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-(-2))**2 + (1-(-3))**2 } ; ; a = sqrt{ 32 } = 5.66 ; ;

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-(-2))**2 + (2-(-3))**2 } ; ; b = sqrt{ 26 } = 5.1 ; ;

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 + (2-1)**2 } ; ; c = sqrt{ 26 } = 5.1 ; ;


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 = 5.66 ; ; b = 5.1 ; ; c = 5.1 ; ;

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

p = a+b+c = 5.66+5.1+5.1 = 15.85 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.85 }{ 2 } = 7.93 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.93 * (7.93-5.66)(7.93-5.1)(7.93-5.1) } ; ; T = sqrt{ 144 } = 12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12 }{ 5.66 } = 4.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12 }{ 5.1 } = 4.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12 }{ 5.1 } = 4.71 ; ;

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{ 5.66**2-5.1**2-5.1**2 }{ 2 * 5.1 * 5.1 } ) = 67° 22'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.1**2-5.66**2-5.1**2 }{ 2 * 5.66 * 5.1 } ) = 56° 18'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.1**2-5.66**2-5.1**2 }{ 2 * 5.1 * 5.66 } ) = 56° 18'36" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12 }{ 7.93 } = 1.51 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.66 }{ 2 * sin 67° 22'49" } = 3.06 ; ;




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