Triangle calculator VC

Please enter the coordinates of the three vertices


Right isosceles triangle.

Sides: a = 8.54440037453   b = 12.08330459736   c = 8.54440037453

Area: T = 36.5
Perimeter: p = 29.17110534642
Semiperimeter: s = 14.58655267321

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

Height: ha = 8.54440037453
Height: hb = 6.04215229868
Height: hc = 8.54440037453

Median: ma = 9.55224865873
Median: mb = 6.04215229868
Median: mc = 9.55224865873

Inradius: r = 2.50224807585
Circumradius: R = 6.04215229868

Vertex coordinates: A[-4; 11] B[4; 8] C[1; 0]
Centroid: CG[0.33333333333; 6.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 2.50224807585]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 90° = 1.57107963268 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 = |BC| = |B-C| ; ; a**2 = (B_x-C_x)**2 + (B_y-C_y)**2 ; ; a = sqrt{ (B_x-C_x)**2 + (B_y-C_y)**2 } ; ; a = sqrt{ (4-1)**2 + (8-0)**2 } ; ; a = sqrt{ 73 } = 8.54 ; ;

2. We compute side b from coordinates using the Pythagorean theorem

b = |AC| = |A-C| ; ; b**2 = (A_x-C_x)**2 + (A_y-C_y)**2 ; ; b = sqrt{ (A_x-C_x)**2 + (A_y-C_y)**2 } ; ; b = sqrt{ (-4-1)**2 + (11-0)**2 } ; ; b = sqrt{ 146 } = 12.08 ; ;

3. We compute side c from coordinates using the Pythagorean theorem

c = |AB| = |A-B| ; ; c**2 = (A_x-B_x)**2 + (A_y-B_y)**2 ; ; c = sqrt{ (A_x-B_x)**2 + (A_y-B_y)**2 } ; ; c = sqrt{ (-4-4)**2 + (11-8)**2 } ; ; c = sqrt{ 73 } = 8.54 ; ;
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 = 8.54 ; ; b = 12.08 ; ; c = 8.54 ; ;

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

p = a+b+c = 8.54+12.08+8.54 = 29.17 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29.17 }{ 2 } = 14.59 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.59 * (14.59-8.54)(14.59-12.08)(14.59-8.54) } ; ; T = sqrt{ 1332.25 } = 36.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 36.5 }{ 8.54 } = 8.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 36.5 }{ 12.08 } = 6.04 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 36.5 }{ 8.54 } = 8.54 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 12.08**2+8.54**2-8.54**2 }{ 2 * 12.08 * 8.54 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.54**2+8.54**2-12.08**2 }{ 2 * 8.54 * 8.54 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 45° - 90° = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 36.5 }{ 14.59 } = 2.5 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.54 }{ 2 * sin 45° } = 6.04 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.08**2+2 * 8.54**2 - 8.54**2 } }{ 2 } = 9.552 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.54**2+2 * 8.54**2 - 12.08**2 } }{ 2 } = 6.042 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.08**2+2 * 8.54**2 - 8.54**2 } }{ 2 } = 9.552 ; ;
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