Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 7.61657731059   b = 7.81102496759   c = 3.60655512755

Area: T = 13.5
Perimeter: p = 19.03215740572
Semiperimeter: s = 9.51657870286

Angle ∠ A = α = 73.49656386182° = 73°29'44″ = 1.28327408797 rad
Angle ∠ B = β = 79.50985229877° = 79°30'31″ = 1.38876855095 rad
Angle ∠ C = γ = 26.99658383941° = 26°59'45″ = 0.47111662643 rad

Height: ha = 3.54552736872
Height: hb = 3.45769957582
Height: hc = 7.4888452649

Median: ma = 4.74334164903
Median: mb = 4.5
Median: mc = 7.5

Inradius: r = 1.41986950548
Circumradius: R = 3.97215207012

Vertex coordinates: A[9; -3] B[11; -6] C[4; -9]
Centroid: CG[8; -6]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.26327213064; 1.41986950548]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 106.5044361382° = 106°30'16″ = 1.28327408797 rad
∠ B' = β' = 100.4911477012° = 100°29'29″ = 1.38876855095 rad
∠ C' = γ' = 153.0044161606° = 153°15″ = 0.47111662643 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{ (11-4)**2 + (-6-(-9))**2 } ; ; a = sqrt{ 58 } = 7.62 ; ;

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{ (9-4)**2 + (-3-(-9))**2 } ; ; b = sqrt{ 61 } = 7.81 ; ;

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{ (9-11)**2 + (-3-(-6))**2 } ; ; c = sqrt{ 13 } = 3.61 ; ;


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 = 7.62 ; ; b = 7.81 ; ; c = 3.61 ; ;

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

p = a+b+c = 7.62+7.81+3.61 = 19.03 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 19.03 }{ 2 } = 9.52 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.52 * (9.52-7.62)(9.52-7.81)(9.52-3.61) } ; ; T = sqrt{ 182.25 } = 13.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13.5 }{ 7.62 } = 3.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13.5 }{ 7.81 } = 3.46 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13.5 }{ 3.61 } = 7.49 ; ;

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{ 7.81**2+3.61**2-7.62**2 }{ 2 * 7.81 * 3.61 } ) = 73° 29'44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.62**2+3.61**2-7.81**2 }{ 2 * 7.62 * 3.61 } ) = 79° 30'31" ; ; gamma = 180° - alpha - beta = 180° - 73° 29'44" - 79° 30'31" = 26° 59'45" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13.5 }{ 9.52 } = 1.42 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.62 }{ 2 * sin 73° 29'44" } = 3.97 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.81**2+2 * 3.61**2 - 7.62**2 } }{ 2 } = 4.743 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.61**2+2 * 7.62**2 - 7.81**2 } }{ 2 } = 4.5 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.81**2+2 * 7.62**2 - 3.61**2 } }{ 2 } = 7.5 ; ;
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