Triangle calculator VC

Please enter the coordinates of the three vertices


Acute isosceles triangle.

Sides: a = 4.24326406871   b = 3.74216573868   c = 3.74216573868

Area: T = 6.53883484153
Perimeter: p = 11.72659554607
Semiperimeter: s = 5.86329777303

Angle ∠ A = α = 69.07551675724° = 69°4'31″ = 1.20655891055 rad
Angle ∠ B = β = 55.46224162138° = 55°27'45″ = 0.9688001774 rad
Angle ∠ C = γ = 55.46224162138° = 55°27'45″ = 0.9688001774 rad

Height: ha = 3.08222070015
Height: hb = 3.49548942351
Height: hc = 3.49548942351

Median: ma = 3.08222070015
Median: mb = 3.53655339059
Median: mc = 3.53655339059

Inradius: r = 1.11551924357
Circumradius: R = 2.27110998958

Vertex coordinates: A[2; -2; 1] B[1; 0; -2] C[0; 1; 2]
Centroid: CG[1; -0.33333333333; 0.33333333333]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 110.9254832428° = 110°55'29″ = 1.20655891055 rad
∠ B' = β' = 124.5387583786° = 124°32'15″ = 0.9688001774 rad
∠ C' = γ' = 124.5387583786° = 124°32'15″ = 0.9688001774 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 + ( beta _z- gamma _z)**2 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 + ( beta _z- gamma _z)**2 } ; ; a = sqrt{ (1-0)**2 + (0-1)**2 + (-2 - 2)**2 } ; ; a = sqrt{ 18 } = 4.24 ; ;

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 + ( alpha _z- gamma _z)**2 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 + ( alpha _z- gamma _z)**2 } ; ; b = sqrt{ (2-0)**2 + (-2-1)**2 + (1 - 2)**2 } ; ; b = sqrt{ 14 } = 3.74 ; ;

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 + ( alpha _z- beta _z)**2 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 + ( alpha _z- beta _z)**2 } ; ; c = sqrt{ (2-1)**2 + (-2-0)**2 + (1 - (-2))**2 } ; ; c = sqrt{ 14 } = 3.74 ; ;


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 = 4.24 ; ; b = 3.74 ; ; c = 3.74 ; ;

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

p = a+b+c = 4.24+3.74+3.74 = 11.73 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.73 }{ 2 } = 5.86 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.86 * (5.86-4.24)(5.86-3.74)(5.86-3.74) } ; ; T = sqrt{ 42.75 } = 6.54 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6.54 }{ 4.24 } = 3.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6.54 }{ 3.74 } = 3.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6.54 }{ 3.74 } = 3.49 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 3.74**2+3.74**2-4.24**2 }{ 2 * 3.74 * 3.74 } ) = 69° 4'31" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.24**2+3.74**2-3.74**2 }{ 2 * 4.24 * 3.74 } ) = 55° 27'45" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 4.24**2+3.74**2-3.74**2 }{ 2 * 4.24 * 3.74 } ) = 55° 27'45" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6.54 }{ 5.86 } = 1.12 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.24 }{ 2 * sin 69° 4'31" } = 2.27 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.74**2+2 * 3.74**2 - 4.24**2 } }{ 2 } = 3.082 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.74**2+2 * 4.24**2 - 3.74**2 } }{ 2 } = 3.536 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.74**2+2 * 4.24**2 - 3.74**2 } }{ 2 } = 3.536 ; ;
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