Triangle calculator VC

Please enter the coordinates of the three vertices


Right isosceles triangle.

Sides: a = 11.66219037897   b = 16.49224225025   c = 11.66219037897

Area: T = 68
Perimeter: p = 39.81662300819
Semiperimeter: s = 19.90881150409

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

Height: ha = 11.66219037897
Height: hb = 8.24662112512
Height: hc = 11.66219037897

Median: ma = 13.03884048104
Median: mb = 8.24662112512
Median: mc = 13.03884048104

Inradius: r = 3.41656925385
Circumradius: R = 8.24662112512

Vertex coordinates: A[6; 3] B[-4; -3] C[-10; 7]
Centroid: CG[-2.66766666667; 2.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 3.41656925385]

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{ (-4-(-10))**2 + (-3-7)**2 } ; ; a = sqrt{ 136 } = 11.66 ; ;

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{ (6-(-10))**2 + (3-7)**2 } ; ; b = sqrt{ 272 } = 16.49 ; ;

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{ (6-(-4))**2 + (3-(-3))**2 } ; ; c = sqrt{ 136 } = 11.66 ; ;


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 = 11.66 ; ; b = 16.49 ; ; c = 11.66 ; ;

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

p = a+b+c = 11.66+16.49+11.66 = 39.82 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 39.82 }{ 2 } = 19.91 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.91 * (19.91-11.66)(19.91-16.49)(19.91-11.66) } ; ; T = sqrt{ 4624 } = 68 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 68 }{ 11.66 } = 11.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 68 }{ 16.49 } = 8.25 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 68 }{ 11.66 } = 11.66 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 16.49**2+11.66**2-11.66**2 }{ 2 * 16.49 * 11.66 } ) = 45° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 11.66**2+11.66**2-16.49**2 }{ 2 * 11.66 * 11.66 } ) = 90° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 11.66**2+16.49**2-11.66**2 }{ 2 * 11.66 * 16.49 } ) = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 68 }{ 19.91 } = 3.42 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.66 }{ 2 * sin 45° } = 8.25 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.49**2+2 * 11.66**2 - 11.66**2 } }{ 2 } = 13.038 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.66**2+2 * 11.66**2 - 16.49**2 } }{ 2 } = 8.246 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.49**2+2 * 11.66**2 - 11.66**2 } }{ 2 } = 13.038 ; ;
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