Triangle calculator VC

Please enter the coordinates of the three vertices


Right isosceles triangle.

Sides: a = 6.70882039325   b = 9.48768329805   c = 6.70882039325

Area: T = 22.5
Perimeter: p = 22.90332408455
Semiperimeter: s = 11.45216204228

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

Height: ha = 6.70882039325
Height: hb = 4.74334164903
Height: hc = 6.70882039325

Median: ma = 7.5
Median: mb = 4.74334164903
Median: mc = 7.5

Inradius: r = 1.96547874422
Circumradius: R = 4.74334164903

Vertex coordinates: A[0; -3] B[-3; 3] C[-9; 0]
Centroid: CG[-4; 0]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 1.96547874422]

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{ (-3-(-9))**2 + (3-0)**2 } ; ; a = sqrt{ 45 } = 6.71 ; ;

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-(-9))**2 + (-3-0)**2 } ; ; b = sqrt{ 90 } = 9.49 ; ;

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{ (0-(-3))**2 + (-3-3)**2 } ; ; c = sqrt{ 45 } = 6.71 ; ;


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 = 6.71 ; ; b = 9.49 ; ; c = 6.71 ; ;

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

p = a+b+c = 6.71+9.49+6.71 = 22.9 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.9 }{ 2 } = 11.45 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.45 * (11.45-6.71)(11.45-9.49)(11.45-6.71) } ; ; T = sqrt{ 506.25 } = 22.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22.5 }{ 6.71 } = 6.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22.5 }{ 9.49 } = 4.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22.5 }{ 6.71 } = 6.71 ; ;

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{ 6.71**2-9.49**2-6.71**2 }{ 2 * 9.49 * 6.71 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.49**2-6.71**2-6.71**2 }{ 2 * 6.71 * 6.71 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.71**2-6.71**2-9.49**2 }{ 2 * 9.49 * 6.71 } ) = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22.5 }{ 11.45 } = 1.96 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.71 }{ 2 * sin 45° } = 4.74 ; ;




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