Triangle calculator VC

Please enter the coordinates of the three vertices


Right isosceles triangle.

Sides: a = 12.16655250606   b = 8.6022325267   c = 8.6022325267

Area: T = 37
Perimeter: p = 29.37701755947
Semiperimeter: s = 14.68550877973

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 6.08327625303
Height: hb = 8.6022325267
Height: hc = 8.6022325267

Median: ma = 6.08327625303
Median: mb = 9.61876920308
Median: mc = 9.61876920308

Inradius: r = 2.52195627367
Circumradius: R = 6.08327625303

Vertex coordinates: A[-10; 4] B[-3; -1] C[-5; 11]
Centroid: CG[-6; 4.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.52195627367; 2.52195627367]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 135° = 0.78553981634 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{ (-3-(-5))**2 + (-1-11)**2 } ; ; a = sqrt{ 148 } = 12.17 ; ;

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{ (-10-(-5))**2 + (4-11)**2 } ; ; b = sqrt{ 74 } = 8.6 ; ;

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{ (-10-(-3))**2 + (4-(-1))**2 } ; ; c = sqrt{ 74 } = 8.6 ; ;


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 = 12.17 ; ; b = 8.6 ; ; c = 8.6 ; ;

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

p = a+b+c = 12.17+8.6+8.6 = 29.37 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29.37 }{ 2 } = 14.69 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.69 * (14.69-12.17)(14.69-8.6)(14.69-8.6) } ; ; T = sqrt{ 1369 } = 37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 37 }{ 12.17 } = 6.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 37 }{ 8.6 } = 8.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 37 }{ 8.6 } = 8.6 ; ;

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{ 12.17**2-8.6**2-8.6**2 }{ 2 * 8.6 * 8.6 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.6**2-12.17**2-8.6**2 }{ 2 * 12.17 * 8.6 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.6**2-12.17**2-8.6**2 }{ 2 * 8.6 * 12.17 } ) = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 37 }{ 14.69 } = 2.52 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.17 }{ 2 * sin 90° } = 6.08 ; ;




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