Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 9.05553851381   b = 8.06222577483   c = 5.38551648071

Area: T = 21.5
Perimeter: p = 22.50328076936
Semiperimeter: s = 11.25114038468

Angle ∠ A = α = 82.05765281894° = 82°3'24″ = 1.43221565897 rad
Angle ∠ B = β = 61.85883987677° = 61°51'30″ = 1.08796327285 rad
Angle ∠ C = γ = 36.08550730429° = 36°5'6″ = 0.63298033354 rad

Height: ha = 4.74985556212
Height: hb = 5.33334935873
Height: hc = 7.98548995416

Median: ma = 5.14878150705
Median: mb = 6.26549820431
Median: mc = 8.1399410298

Inradius: r = 1.91108726602
Circumradius: R = 4.57215571382

Vertex coordinates: A[0; 6] B[5; 8] C[4; -1]
Centroid: CG[3; 4.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.02220946787; 1.91108726602]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 97.94334718106° = 97°56'36″ = 1.43221565897 rad
∠ B' = β' = 118.1421601232° = 118°8'30″ = 1.08796327285 rad
∠ C' = γ' = 143.9154926957° = 143°54'54″ = 0.63298033354 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{ (5-4)**2 + (8-(-1))**2 } ; ; a = sqrt{ 82 } = 9.06 ; ;

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{ (0-4)**2 + (6-(-1))**2 } ; ; b = sqrt{ 65 } = 8.06 ; ;

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{ (0-5)**2 + (6-8)**2 } ; ; c = sqrt{ 29 } = 5.39 ; ;


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 = 9.06 ; ; b = 8.06 ; ; c = 5.39 ; ;

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

p = a+b+c = 9.06+8.06+5.39 = 22.5 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.5 }{ 2 } = 11.25 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.25 * (11.25-9.06)(11.25-8.06)(11.25-5.39) } ; ; T = sqrt{ 462.25 } = 21.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21.5 }{ 9.06 } = 4.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21.5 }{ 8.06 } = 5.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21.5 }{ 5.39 } = 7.98 ; ;

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{ 9.06**2-8.06**2-5.39**2 }{ 2 * 8.06 * 5.39 } ) = 82° 3'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.06**2-9.06**2-5.39**2 }{ 2 * 9.06 * 5.39 } ) = 61° 51'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.39**2-9.06**2-8.06**2 }{ 2 * 8.06 * 9.06 } ) = 36° 5'6" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21.5 }{ 11.25 } = 1.91 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.06 }{ 2 * sin 82° 3'24" } = 4.57 ; ;




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