Triangle calculator VC

Please enter the coordinates of the three vertices


Acute isosceles triangle.

Sides: a = 7.07110678119   b = 4.4722135955   c = 7.07110678119

Area: T = 15
Perimeter: p = 18.61442715787
Semiperimeter: s = 9.30771357894

Angle ∠ A = α = 71.56550511771° = 71°33'54″ = 1.24990457724 rad
Angle ∠ B = β = 36.87698976458° = 36°52'12″ = 0.64435011088 rad
Angle ∠ C = γ = 71.56550511771° = 71°33'54″ = 1.24990457724 rad

Height: ha = 4.24326406871
Height: hb = 6.70882039325
Height: hc = 4.24326406871

Median: ma = 4.74334164903
Median: mb = 6.70882039325
Median: mc = 4.74334164903

Inradius: r = 1.61216666115
Circumradius: R = 3.72767799625

Vertex coordinates: A[4; -1] B[-1; 4] C[0; -3]
Centroid: CG[1; 0]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.14988888153; 1.61216666115]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108.4354948823° = 108°26'6″ = 1.24990457724 rad
∠ B' = β' = 143.1330102354° = 143°7'48″ = 0.64435011088 rad
∠ C' = γ' = 108.4354948823° = 108°26'6″ = 1.24990457724 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{ (-1-0)**2 + (4-(-3))**2 } ; ; a = sqrt{ 50 } = 7.07 ; ;

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{ (4-0)**2 + (-1-(-3))**2 } ; ; b = sqrt{ 20 } = 4.47 ; ;

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{ (4-(-1))**2 + (-1-4)**2 } ; ; c = sqrt{ 50 } = 7.07 ; ;


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 = 7.07 ; ; b = 4.47 ; ; c = 7.07 ; ;

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

p = a+b+c = 7.07+4.47+7.07 = 18.61 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.61 }{ 2 } = 9.31 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.31 * (9.31-7.07)(9.31-4.47)(9.31-7.07) } ; ; T = sqrt{ 225 } = 15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 15 }{ 7.07 } = 4.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 15 }{ 4.47 } = 6.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 15 }{ 7.07 } = 4.24 ; ;

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{ 7.07**2-4.47**2-7.07**2 }{ 2 * 4.47 * 7.07 } ) = 71° 33'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.47**2-7.07**2-7.07**2 }{ 2 * 7.07 * 7.07 } ) = 36° 52'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.07**2-7.07**2-4.47**2 }{ 2 * 4.47 * 7.07 } ) = 71° 33'54" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 15 }{ 9.31 } = 1.61 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.07 }{ 2 * sin 71° 33'54" } = 3.73 ; ;




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