Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 9.84988578018   b = 10.81766538264   c = 5.83109518948

Area: T = 28.5
Perimeter: p = 26.4966463523
Semiperimeter: s = 13.24882317615

Angle ∠ A = α = 64.65438240581° = 64°39'14″ = 1.12884221038 rad
Angle ∠ B = β = 82.99987324425° = 82°59'55″ = 1.44986011561 rad
Angle ∠ C = γ = 32.34774434994° = 32°20'51″ = 0.56545693937 rad

Height: ha = 5.78774731413
Height: hb = 5.27696518641
Height: hc = 9.77554193531

Median: ma = 7.15989105316
Median: mb = 6.02107972894
Median: mc = 9.92547166206

Inradius: r = 2.15112304822
Circumradius: R = 5.44989573071

Vertex coordinates: A[5; 4] B[0; 7] C[-4; -2]
Centroid: CG[0.33333333333; 3]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.26441861996; 2.15112304822]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115.3466175942° = 115°20'46″ = 1.12884221038 rad
∠ B' = β' = 97.00112675575° = 97°5″ = 1.44986011561 rad
∠ C' = γ' = 147.6532556501° = 147°39'9″ = 0.56545693937 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{ (0-(-4))**2 + (7-(-2))**2 } ; ; a = sqrt{ 97 } = 9.85 ; ;

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{ (5-(-4))**2 + (4-(-2))**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{ (5-0)**2 + (4-7)**2 } ; ; c = sqrt{ 34 } = 5.83 ; ;


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 = 9.85 ; ; b = 10.82 ; ; c = 5.83 ; ;

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

p = a+b+c = 9.85+10.82+5.83 = 26.5 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.5 }{ 2 } = 13.25 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.25 * (13.25-9.85)(13.25-10.82)(13.25-5.83) } ; ; T = sqrt{ 812.25 } = 28.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 28.5 }{ 9.85 } = 5.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 28.5 }{ 10.82 } = 5.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 28.5 }{ 5.83 } = 9.78 ; ;

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{ 9.85**2-10.82**2-5.83**2 }{ 2 * 10.82 * 5.83 } ) = 64° 39'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10.82**2-9.85**2-5.83**2 }{ 2 * 9.85 * 5.83 } ) = 82° 59'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.83**2-9.85**2-10.82**2 }{ 2 * 10.82 * 9.85 } ) = 32° 20'51" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 28.5 }{ 13.25 } = 2.15 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.85 }{ 2 * sin 64° 39'14" } = 5.45 ; ;




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