Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 3.60655512755   b = 7.07110678119   c = 6.40331242374

Area: T = 11.5
Perimeter: p = 17.08797433248
Semiperimeter: s = 8.54398716624

Angle ∠ A = α = 30.53297058999° = 30°31'47″ = 0.53328438876 rad
Angle ∠ B = β = 85.03302592719° = 85°1'49″ = 1.48440579881 rad
Angle ∠ C = γ = 64.44400348282° = 64°26'24″ = 1.12546907779 rad

Height: ha = 6.37990522566
Height: hb = 3.25326911935
Height: hc = 3.59219965234

Median: ma = 6.5
Median: mb = 3.80878865529
Median: mc = 4.61097722286

Inradius: r = 1.34766244523
Circumradius: R = 3.54988755907

Vertex coordinates: A[0; 0] B[4; 5] C[1; 7]
Centroid: CG[1.66766666667; 4]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.11770977785; 1.34766244523]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.47702941° = 149°28'13″ = 0.53328438876 rad
∠ B' = β' = 94.97697407281° = 94°58'11″ = 1.48440579881 rad
∠ C' = γ' = 115.5659965172° = 115°33'36″ = 1.12546907779 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{ (4-1)**2 + (5-7)**2 } ; ; a = sqrt{ 13 } = 3.61 ; ;

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{ (0-1)**2 + (0-7)**2 } ; ; b = sqrt{ 50 } = 7.07 ; ;

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{ (0-4)**2 + (0-5)**2 } ; ; c = sqrt{ 41 } = 6.4 ; ;


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 = 3.61 ; ; b = 7.07 ; ; c = 6.4 ; ;

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

p = a+b+c = 3.61+7.07+6.4 = 17.08 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.08 }{ 2 } = 8.54 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.54 * (8.54-3.61)(8.54-7.07)(8.54-6.4) } ; ; T = sqrt{ 132.25 } = 11.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11.5 }{ 3.61 } = 6.38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11.5 }{ 7.07 } = 3.25 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11.5 }{ 6.4 } = 3.59 ; ;

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{ 3.61**2-7.07**2-6.4**2 }{ 2 * 7.07 * 6.4 } ) = 30° 31'47" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.07**2-3.61**2-6.4**2 }{ 2 * 3.61 * 6.4 } ) = 85° 1'49" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.4**2-3.61**2-7.07**2 }{ 2 * 7.07 * 3.61 } ) = 64° 26'24" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11.5 }{ 8.54 } = 1.35 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.61 }{ 2 * sin 30° 31'47" } = 3.55 ; ;




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