Triangle calculator VC

Please enter the coordinates of the three vertices


Acute isosceles triangle.

Sides: a = 2.44994897428   b = 3.74216573868   c = 3.74216573868

Area: T = 4.33301270189
Perimeter: p = 9.93328045163
Semiperimeter: s = 4.96664022582

Angle ∠ A = α = 38.21332107017° = 38°12'48″ = 0.66769463445 rad
Angle ∠ B = β = 70.89333946491° = 70°53'36″ = 1.23773231545 rad
Angle ∠ C = γ = 70.89333946491° = 70°53'36″ = 1.23773231545 rad

Height: ha = 3.53655339059
Height: hb = 2.31545502494
Height: hc = 2.31545502494

Median: ma = 3.53655339059
Median: mb = 2.55495097568
Median: mc = 2.55495097568

Inradius: r = 0.8721884071
Circumradius: R = 1.98798989873

Vertex coordinates: A[-2; 0; -1] B[-1; 2; 2] C[1; 1; 1]
Centroid: CG[-0.66766666667; 1; 0.66766666667]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.7876789298° = 141°47'12″ = 0.66769463445 rad
∠ B' = β' = 109.1076605351° = 109°6'24″ = 1.23773231545 rad
∠ C' = γ' = 109.1076605351° = 109°6'24″ = 1.23773231545 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{ (-1-1)**2 + (2-1)**2 + (2 - 1)**2 } ; ; a = sqrt{ 6 } = 2.45 ; ;

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{ (-2-1)**2 + (0-1)**2 + (-1 - 1)**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 + (0-2)**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 = 2.45 ; ; b = 3.74 ; ; c = 3.74 ; ;

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

p = a+b+c = 2.45+3.74+3.74 = 9.93 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 9.93 }{ 2 } = 4.97 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.97 * (4.97-2.45)(4.97-3.74)(4.97-3.74) } ; ; T = sqrt{ 18.75 } = 4.33 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.33 }{ 2.45 } = 3.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.33 }{ 3.74 } = 2.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.33 }{ 3.74 } = 2.31 ; ;

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{ 2.45**2-3.74**2-3.74**2 }{ 2 * 3.74 * 3.74 } ) = 38° 12'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.74**2-2.45**2-3.74**2 }{ 2 * 2.45 * 3.74 } ) = 70° 53'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.74**2-2.45**2-3.74**2 }{ 2 * 3.74 * 2.45 } ) = 70° 53'36" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.33 }{ 4.97 } = 0.87 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2.45 }{ 2 * sin 38° 12'48" } = 1.98 ; ;




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