Triangle calculator VC

Please enter the coordinates of the three vertices


Right isosceles triangle.

Sides: a = 10.77703296143   b = 15.23215462117   c = 10.77703296143

Area: T = 58
Perimeter: p = 36.77222054403
Semiperimeter: s = 18.38661027201

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 10.77703296143
Height: hb = 7.61657731059
Height: hc = 10.77703296143

Median: ma = 12.04215945788
Median: mb = 7.61657731059
Median: mc = 12.04215945788

Inradius: r = 3.15545565084
Circumradius: R = 7.61657731059

Vertex coordinates: A[9; -3] B[-1; 1] C[-5; -9]
Centroid: CG[1; -3.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 3.15545565084]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 135° = 0.78553981634 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-(-5))**2 + (1-(-9))**2 } ; ; a = sqrt{ 116 } = 10.77 ; ;

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-(-5))**2 + (-3-(-9))**2 } ; ; b = sqrt{ 232 } = 15.23 ; ;

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-(-1))**2 + (-3-1)**2 } ; ; c = sqrt{ 116 } = 10.77 ; ;


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 = 10.77 ; ; b = 15.23 ; ; c = 10.77 ; ;

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

p = a+b+c = 10.77+15.23+10.77 = 36.77 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 36.77 }{ 2 } = 18.39 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.39 * (18.39-10.77)(18.39-15.23)(18.39-10.77) } ; ; T = sqrt{ 3364 } = 58 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 58 }{ 10.77 } = 10.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 58 }{ 15.23 } = 7.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 58 }{ 10.77 } = 10.77 ; ;

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{ 10.77**2-15.23**2-10.77**2 }{ 2 * 15.23 * 10.77 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15.23**2-10.77**2-10.77**2 }{ 2 * 10.77 * 10.77 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.77**2-10.77**2-15.23**2 }{ 2 * 15.23 * 10.77 } ) = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 58 }{ 18.39 } = 3.15 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.77 }{ 2 * sin 45° } = 7.62 ; ;




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