Triangle calculator VC

Please enter the coordinates of the three vertices


Right isosceles triangle.

Sides: a = 10.81766538264   b = 10.81766538264   c = 15.29770585408

Area: T = 58.5
Perimeter: p = 36.93303661936
Semiperimeter: s = 18.46551830968

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

Height: ha = 10.81766538264
Height: hb = 10.81766538264
Height: hc = 7.64985292704

Median: ma = 12.09333866224
Median: mb = 12.09333866224
Median: mc = 7.64985292704

Inradius: r = 3.1688124556
Circumradius: R = 7.64985292704

Vertex coordinates: A[10; 6] B[7; -9] C[1; 0]
Centroid: CG[6; -1]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[3.1688124556; 3.1688124556]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 90° = 1.57107963268 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{ (7-1)**2 + (-9-0)**2 } ; ; a = sqrt{ 117 } = 10.82 ; ;

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{ (10-1)**2 + (6-0)**2 } ; ; b = sqrt{ 117 } = 10.82 ; ;

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{ (10-7)**2 + (6-(-9))**2 } ; ; c = sqrt{ 234 } = 15.3 ; ;


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 = 10.82 ; ; b = 10.82 ; ; c = 15.3 ; ;

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

p = a+b+c = 10.82+10.82+15.3 = 36.93 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 36.93 }{ 2 } = 18.47 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.47 * (18.47-10.82)(18.47-10.82)(18.47-15.3) } ; ; T = sqrt{ 3422.25 } = 58.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 58.5 }{ 10.82 } = 10.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 58.5 }{ 10.82 } = 10.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 58.5 }{ 15.3 } = 7.65 ; ;

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{ 10.82**2-10.82**2-15.3**2 }{ 2 * 10.82 * 15.3 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10.82**2-10.82**2-15.3**2 }{ 2 * 10.82 * 15.3 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15.3**2-10.82**2-10.82**2 }{ 2 * 10.82 * 10.82 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 58.5 }{ 18.47 } = 3.17 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.82 }{ 2 * sin 45° } = 7.65 ; ;




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