Triangle calculator VC

Please enter the coordinates of the three vertices


Acute isosceles triangle.

Sides: a = 3.46441016151   b = 3.31766247904   c = 3.31766247904

Area: T = 4.89989794856
Perimeter: p = 10.09773511958
Semiperimeter: s = 5.04986755979

Angle ∠ A = α = 62.96443082106° = 62°57'52″ = 1.09989344895 rad
Angle ∠ B = β = 58.51878458947° = 58°31'4″ = 1.0211329082 rad
Angle ∠ C = γ = 58.51878458947° = 58°31'4″ = 1.0211329082 rad

Height: ha = 2.82884271247
Height: hb = 2.95441957835
Height: hc = 2.95441957835

Median: ma = 2.82884271247
Median: mb = 2.95880398915
Median: mc = 2.95880398915

Inradius: r = 0.97703494294
Circumradius: R = 1.94545436483

Vertex coordinates: A[1; -1; -2] B[-2; 0; -1] C[0; -2; 1]
Centroid: CG[-0.33333333333; -1; -0.66766666667]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 117.0365691789° = 117°2'8″ = 1.09989344895 rad
∠ B' = β' = 121.4822154105° = 121°28'56″ = 1.0211329082 rad
∠ C' = γ' = 121.4822154105° = 121°28'56″ = 1.0211329082 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{ (-2-0)**2 + (0-(-2))**2 + (-1 - 1)**2 } ; ; a = sqrt{ 12 } = 3.46 ; ;

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{ (1-0)**2 + (-1-(-2))**2 + (-2 - 1)**2 } ; ; b = sqrt{ 11 } = 3.32 ; ;

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{ (1-(-2))**2 + (-1-0)**2 + (-2 - (-1))**2 } ; ; c = sqrt{ 11 } = 3.32 ; ;


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 = 3.46 ; ; b = 3.32 ; ; c = 3.32 ; ;

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

p = a+b+c = 3.46+3.32+3.32 = 10.1 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 10.1 }{ 2 } = 5.05 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.05 * (5.05-3.46)(5.05-3.32)(5.05-3.32) } ; ; T = sqrt{ 24 } = 4.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.9 }{ 3.46 } = 2.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.9 }{ 3.32 } = 2.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.9 }{ 3.32 } = 2.95 ; ;

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{ 3.46**2-3.32**2-3.32**2 }{ 2 * 3.32 * 3.32 } ) = 62° 57'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.32**2-3.46**2-3.32**2 }{ 2 * 3.46 * 3.32 } ) = 58° 31'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.32**2-3.46**2-3.32**2 }{ 2 * 3.32 * 3.46 } ) = 58° 31'4" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.9 }{ 5.05 } = 0.97 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.46 }{ 2 * sin 62° 57'52" } = 1.94 ; ;




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