Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 11.4021754251   b = 9.48768329805   c = 6.32545553203

Area: T = 30
Perimeter: p = 27.21331425518
Semiperimeter: s = 13.60765712759

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 56.3109932474° = 56°18'36″ = 0.98327937232 rad
Angle ∠ C = γ = 33.6990067526° = 33°41'24″ = 0.58880026035 rad

Height: ha = 5.26223481158
Height: hb = 6.32545553203
Height: hc = 9.48768329805

Median: ma = 5.70108771255
Median: mb = 7.90656941504
Median: mc = 10

Inradius: r = 2.20548170249
Circumradius: R = 5.70108771255

Vertex coordinates: A[-4; 3] B[2; 5] C[-1; -6]
Centroid: CG[-1; 0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.47698780166; 2.20548170249]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 123.6990067526° = 123°41'24″ = 0.98327937232 rad
∠ C' = γ' = 146.3109932474° = 146°18'36″ = 0.58880026035 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{ (2-(-1))**2 + (5-(-6))**2 } ; ; a = sqrt{ 130 } = 11.4 ; ;

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{ (-4-(-1))**2 + (3-(-6))**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{ (-4-2)**2 + (3-5)**2 } ; ; c = sqrt{ 40 } = 6.32 ; ;


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 = 11.4 ; ; b = 9.49 ; ; c = 6.32 ; ;

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

p = a+b+c = 11.4+9.49+6.32 = 27.21 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.21 }{ 2 } = 13.61 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.61 * (13.61-11.4)(13.61-9.49)(13.61-6.32) } ; ; T = sqrt{ 900 } = 30 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 30 }{ 11.4 } = 5.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 30 }{ 9.49 } = 6.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 30 }{ 6.32 } = 9.49 ; ;

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{ 11.4**2-9.49**2-6.32**2 }{ 2 * 9.49 * 6.32 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.49**2-11.4**2-6.32**2 }{ 2 * 11.4 * 6.32 } ) = 56° 18'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.32**2-11.4**2-9.49**2 }{ 2 * 9.49 * 11.4 } ) = 33° 41'24" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 30 }{ 13.61 } = 2.2 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.4 }{ 2 * sin 90° } = 5.7 ; ;




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