Triangle calculator VC

Please enter the coordinates of the three vertices


Acute isosceles triangle.

Sides: a = 8.06222577483   b = 8.06222577483   c = 7.21111025509

Area: T = 26
Perimeter: p = 23.33656180475
Semiperimeter: s = 11.66878090238

Angle ∠ A = α = 63.43549488229° = 63°26'6″ = 1.10771487178 rad
Angle ∠ B = β = 63.43549488229° = 63°26'6″ = 1.10771487178 rad
Angle ∠ C = γ = 53.13301023542° = 53°7'48″ = 0.9277295218 rad

Height: ha = 6.45498061986
Height: hb = 6.45498061986
Height: hc = 7.21111025509

Median: ma = 6.5
Median: mb = 6.5
Median: mc = 7.21111025509

Inradius: r = 2.22883532364
Circumradius: R = 4.50769390943

Vertex coordinates: A[-9; 6] B[-3; 10] C[-2; 2]
Centroid: CG[-4.66766666667; 6]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.11441766182; 2.22883532364]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.5655051177° = 116°33'54″ = 1.10771487178 rad
∠ B' = β' = 116.5655051177° = 116°33'54″ = 1.10771487178 rad
∠ C' = γ' = 126.8769897646° = 126°52'12″ = 0.9277295218 rad

Calculate another triangle




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{ (-3-(-2))**2 + (10-2)**2 } ; ; a = sqrt{ 65 } = 8.06 ; ;

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{ (-9-(-2))**2 + (6-2)**2 } ; ; b = sqrt{ 65 } = 8.06 ; ;

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{ (-9-(-3))**2 + (6-10)**2 } ; ; c = sqrt{ 52 } = 7.21 ; ;


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 = 8.06 ; ; b = 8.06 ; ; c = 7.21 ; ;

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

p = a+b+c = 8.06+8.06+7.21 = 23.34 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.34 }{ 2 } = 11.67 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.67 * (11.67-8.06)(11.67-8.06)(11.67-7.21) } ; ; T = sqrt{ 676 } = 26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 26 }{ 8.06 } = 6.45 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 26 }{ 8.06 } = 6.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 26 }{ 7.21 } = 7.21 ; ;

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{ 8.06**2-8.06**2-7.21**2 }{ 2 * 8.06 * 7.21 } ) = 63° 26'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.06**2-8.06**2-7.21**2 }{ 2 * 8.06 * 7.21 } ) = 63° 26'6" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.21**2-8.06**2-8.06**2 }{ 2 * 8.06 * 8.06 } ) = 53° 7'48" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 26 }{ 11.67 } = 2.23 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.06 }{ 2 * sin 63° 26'6" } = 4.51 ; ;




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