Triangle calculator VC

Please enter the coordinates of the three vertices


Acute isosceles triangle.

Sides: a = 1.41442135624   b = 2.23660679775   c = 2.23660679775

Area: T = 1.5
Perimeter: p = 5.88663495174
Semiperimeter: s = 2.94331747587

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

Height: ha = 2.12113203436
Height: hb = 1.34216407865
Height: hc = 1.34216407865

Median: ma = 2.12113203436
Median: mb = 1.5
Median: mc = 1.5

Inradius: r = 0.51096537321
Circumradius: R = 1.1798511302

Vertex coordinates: A[0; 0] B[1; 2] C[2; 1]
Centroid: CG[1; 1]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.17698845774; 0.51096537321]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.1330102354° = 143°7'48″ = 0.64435011088 rad
∠ B' = β' = 108.4354948823° = 108°26'6″ = 1.24990457724 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{ (1-2)**2 + (2-1)**2 } ; ; a = sqrt{ 2 } = 1.41 ; ;

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{ (0-2)**2 + (0-1)**2 } ; ; b = sqrt{ 5 } = 2.24 ; ;

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{ (0-1)**2 + (0-2)**2 } ; ; c = sqrt{ 5 } = 2.24 ; ;


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 = 1.41 ; ; b = 2.24 ; ; c = 2.24 ; ;

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

p = a+b+c = 1.41+2.24+2.24 = 5.89 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 5.89 }{ 2 } = 2.94 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.94 * (2.94-1.41)(2.94-2.24)(2.94-2.24) } ; ; T = sqrt{ 2.25 } = 1.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.5 }{ 1.41 } = 2.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.5 }{ 2.24 } = 1.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.5 }{ 2.24 } = 1.34 ; ;

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{ 1.41**2-2.24**2-2.24**2 }{ 2 * 2.24 * 2.24 } ) = 36° 52'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2.24**2-1.41**2-2.24**2 }{ 2 * 1.41 * 2.24 } ) = 71° 33'54" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.24**2-1.41**2-2.24**2 }{ 2 * 2.24 * 1.41 } ) = 71° 33'54" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.5 }{ 2.94 } = 0.51 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.41 }{ 2 * sin 36° 52'12" } = 1.18 ; ;




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