Triangle calculator - result

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, c and angle γ.

Right scalene Pythagorean triangle.

Sides: a = 3   b = 4   c = 5

Area: T = 6
Perimeter: p = 12
Semiperimeter: s = 6

Angle ∠ A = α = 36.87698976458° = 36°52'12″ = 0.64435011088 rad
Angle ∠ B = β = 53.13301023542° = 53°7'48″ = 0.9277295218 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 4
Height: hb = 3
Height: hc = 2.4

Median: ma = 4.27220018727
Median: mb = 3.60655512755
Median: mc = 2.5

Inradius: r = 1
Circumradius: R = 2.5

Vertex coordinates: A[5; 0] B[0; 0] C[1.8; 2.4]
Centroid: CG[2.26766666667; 0.8]
Coordinates of the circumscribed circle: U[2.5; 0]
Coordinates of the inscribed circle: I[2; 1]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.1330102354° = 143°7'48″ = 0.64435011088 rad
∠ B' = β' = 126.8769897646° = 126°52'12″ = 0.9277295218 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: side a, c and angle γ.

a = 3 ; ; c = 5 ; ; gamma = 90° ; ;

2. From angle γ, side a and side c we calculate side b - by using the law of cosines and quadratic equation:

c**2 = a**2 + b**2 - 2a b cos gamma ; ; ; ; 5**2 = 3**2 + b**2 - 2 * 3 * b * cos 90° ; ; ; ; ; ; b**2 -16 =0 ; ; p=1; q=-0; r=-16 ; ; D = q**2 - 4pr = 0**2 - 4 * 1 * (-16) = 64 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ ± sqrt{ 64 } }{ 2 } ; ; b_{1,2} = fraction{ ± 8 }{ 2 } ; ; b_{1,2} = ± 4 ; ; b_{1} = 4 ; ; b_{2} = -4 ; ; ; ; text{ Factored form: } ; ; (b -4) (b +4) = 0 ; ; ; ; b > 0 ; ; ; ; b = 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 = 3 ; ; b = 4 ; ; c = 5 ; ;

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

p = a+b+c = 3+4+5 = 12 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12 }{ 2 } = 6 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6 * (6-3)(6-4)(6-5) } ; ; T = sqrt{ 36 } = 6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6 }{ 3 } = 4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6 }{ 4 } = 3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6 }{ 5 } = 2.4 ; ;

7. 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{ 4**2+5**2-3**2 }{ 2 * 4 * 5 } ) = 36° 52'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3**2+5**2-4**2 }{ 2 * 3 * 5 } ) = 53° 7'48" ; ; gamma = 180° - alpha - beta = 180° - 36° 52'12" - 53° 7'48" = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6 }{ 6 } = 1 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3 }{ 2 * sin 36° 52'12" } = 2.5 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4**2+2 * 5**2 - 3**2 } }{ 2 } = 4.272 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5**2+2 * 3**2 - 4**2 } }{ 2 } = 3.606 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4**2+2 * 3**2 - 5**2 } }{ 2 } = 2.5 ; ;
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