Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 15.52441746963   b = 15.62204993518   c = 16.76330546142

Area: T = 110
Perimeter: p = 47.90877286623
Semiperimeter: s = 23.95438643312

Angle ∠ A = α = 57.16595957285° = 57°9'35″ = 0.99876231446 rad
Angle ∠ B = β = 57.71545581856° = 57°42'52″ = 1.00773090667 rad
Angle ∠ C = γ = 65.12658460859° = 65°7'33″ = 1.13766604423 rad

Height: ha = 14.17114457808
Height: hb = 14.08440567926
Height: hc = 13.1244099698

Median: ma = 14.22114626533
Median: mb = 14.14221356237
Median: mc = 13.12444047484

Inradius: r = 4.5922160934
Circumradius: R = 9.23985522192

Vertex coordinates: A[3; -6] B[8; 10] C[-7; 6]
Centroid: CG[1.33333333333; 3.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.90114107719; 4.5922160934]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 122.8440404271° = 122°50'25″ = 0.99876231446 rad
∠ B' = β' = 122.2855441814° = 122°17'8″ = 1.00773090667 rad
∠ C' = γ' = 114.8744153914° = 114°52'27″ = 1.13766604423 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{ (8-(-7))**2 + (10-6)**2 } ; ; a = sqrt{ 241 } = 15.52 ; ;

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{ (3-(-7))**2 + (-6-6)**2 } ; ; b = sqrt{ 244 } = 15.62 ; ;

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{ (3-8)**2 + (-6-10)**2 } ; ; c = sqrt{ 281 } = 16.76 ; ;


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 = 15.52 ; ; b = 15.62 ; ; c = 16.76 ; ;

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

p = a+b+c = 15.52+15.62+16.76 = 47.91 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47.91 }{ 2 } = 23.95 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.95 * (23.95-15.52)(23.95-15.62)(23.95-16.76) } ; ; T = sqrt{ 12100 } = 110 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 110 }{ 15.52 } = 14.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 110 }{ 15.62 } = 14.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 110 }{ 16.76 } = 13.12 ; ;

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{ 15.52**2-15.62**2-16.76**2 }{ 2 * 15.62 * 16.76 } ) = 57° 9'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15.62**2-15.52**2-16.76**2 }{ 2 * 15.52 * 16.76 } ) = 57° 42'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16.76**2-15.52**2-15.62**2 }{ 2 * 15.62 * 15.52 } ) = 65° 7'33" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 110 }{ 23.95 } = 4.59 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15.52 }{ 2 * sin 57° 9'35" } = 9.24 ; ;




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