Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 20.16   b = 2.7   c = 20.34

Area: T = 27.216
Perimeter: p = 43.2
Semiperimeter: s = 21.6

Angle ∠ A = α = 82.37218503314° = 82°22'19″ = 1.43876599992 rad
Angle ∠ B = β = 7.62881496686° = 7°37'41″ = 0.13331363276 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 2.7
Height: hb = 20.16
Height: hc = 2.67661061947

Median: ma = 10.43553437893
Median: mb = 20.20551503335
Median: mc = 10.17

Inradius: r = 1.26
Circumradius: R = 10.17

Vertex coordinates: A[20.16; 0] B[0; 2.7] C[20.16; 2.7]
Centroid: CG[13.44; 1.8]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[9.408; 1.26]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 97.62881496686° = 97°37'41″ = 1.43876599992 rad
∠ B' = β' = 172.3721850331° = 172°22'19″ = 0.13331363276 rad
∠ C' = γ' = 90° = 1.57107963268 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{ (0-20.16)**2 + (2.7-2.7)**2 } ; ; a = sqrt{ 406.426 } = 20.16 ; ;

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{ (20.16-20.16)**2 + (0-2.7)**2 } ; ; b = sqrt{ 7.29 } = 2.7 ; ;

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{ (20.16-0)**2 + (0-2.7)**2 } ; ; c = sqrt{ 413.716 } = 20.34 ; ;


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 = 20.16 ; ; b = 2.7 ; ; c = 20.34 ; ;

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

p = a+b+c = 20.16+2.7+20.34 = 43.2 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43.2 }{ 2 } = 21.6 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.6 * (21.6-20.16)(21.6-2.7)(21.6-20.34) } ; ; T = sqrt{ 740.71 } = 27.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 27.22 }{ 20.16 } = 2.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 27.22 }{ 2.7 } = 20.16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 27.22 }{ 20.34 } = 2.68 ; ;

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{ 20.16**2-2.7**2-20.34**2 }{ 2 * 2.7 * 20.34 } ) = 82° 22'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2.7**2-20.16**2-20.34**2 }{ 2 * 20.16 * 20.34 } ) = 7° 37'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20.34**2-20.16**2-2.7**2 }{ 2 * 2.7 * 20.16 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 27.22 }{ 21.6 } = 1.26 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20.16 }{ 2 * sin 82° 22'19" } = 10.17 ; ;




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