Triangle calculator VC

Please enter the coordinates of the three vertices


Right isosceles triangle.

Sides: a = 6.40331242374   b = 9.05553851381   c = 6.40331242374

Area: T = 20.5
Perimeter: p = 21.8621633613
Semiperimeter: s = 10.93108168065

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

Height: ha = 6.40331242374
Height: hb = 4.52876925691
Height: hc = 6.40331242374

Median: ma = 7.15989105316
Median: mb = 4.52876925691
Median: mc = 7.15989105316

Inradius: r = 1.87554316684
Circumradius: R = 4.52876925691

Vertex coordinates: A[-2; 9] B[3; 5] C[-1; 0]
Centroid: CG[0; 4.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 1.87554316684]

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{ (3-(-1))**2 + (5-0)**2 } ; ; a = sqrt{ 41 } = 6.4 ; ;

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{ (-2-(-1))**2 + (9-0)**2 } ; ; b = sqrt{ 82 } = 9.06 ; ;

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{ (-2-3)**2 + (9-5)**2 } ; ; c = sqrt{ 41 } = 6.4 ; ;


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 = 6.4 ; ; b = 9.06 ; ; c = 6.4 ; ;

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

p = a+b+c = 6.4+9.06+6.4 = 21.86 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.86 }{ 2 } = 10.93 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.93 * (10.93-6.4)(10.93-9.06)(10.93-6.4) } ; ; T = sqrt{ 420.25 } = 20.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20.5 }{ 6.4 } = 6.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20.5 }{ 9.06 } = 4.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20.5 }{ 6.4 } = 6.4 ; ;

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{ 9.06**2+6.4**2-6.4**2 }{ 2 * 9.06 * 6.4 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.4**2+6.4**2-9.06**2 }{ 2 * 6.4 * 6.4 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 45° - 90° = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20.5 }{ 10.93 } = 1.88 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.4 }{ 2 * sin 45° } = 4.53 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.06**2+2 * 6.4**2 - 6.4**2 } }{ 2 } = 7.159 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.4**2+2 * 6.4**2 - 9.06**2 } }{ 2 } = 4.528 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.06**2+2 * 6.4**2 - 6.4**2 } }{ 2 } = 7.159 ; ;
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