Triangle calculator - result

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

You have entered side a, area S and angle β.

Obtuse scalene triangle.

Sides: a = 3   b = 16.26601938325   c = 18.93296126652

Area: T = 12
Perimeter: p = 38.19898064977
Semiperimeter: s = 19.09549032489

Angle ∠ A = α = 4.4722058555° = 4°28'19″ = 0.07880521461 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 150.5287941445° = 150°31'41″ = 2.62772081945 rad

Height: ha = 8
Height: hb = 1.47659971651
Height: hc = 1.26878547852

Median: ma = 17.58215832496
Median: mb = 10.8432815223
Median: mc = 6.86439924841

Inradius: r = 0.62884399477
Circumradius: R = 19.23774481944

Vertex coordinates: A[18.93296126652; 0] B[0; 0] C[2.71989233611; 1.26878547852]
Centroid: CG[7.21661786754; 0.42326182617]
Coordinates of the circumscribed circle: U[9.46548063326; -16.74880403068]
Coordinates of the inscribed circle: I[2.83547094164; 0.62884399477]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 175.5287941445° = 175°31'41″ = 0.07880521461 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 29.4722058555° = 29°28'19″ = 2.62772081945 rad

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How did we calculate this triangle?

1. Input data entered: side a, angle β and area S.

a = 3 ; ; beta = 25° ; ; S = 12 ; ;

2. From side a and angle β we calculate height hc:

h_c = a * sin beta = 3 * sin 25° = 1.268 ; ;

3. From area T and side a we calculate height ha - The area of the triangle is the product of the length of the base and the height divided by two:

T = fraction{ a h_a }{ 2 } ; ; ; ; h_a = fraction{ 2 T }{ a } = fraction{ 2 * 12 }{ 3 } = 8 ; ;

4. From area T, side a and angle β we calculate side c:

T = fraction{ a c sin beta }{ 2 } ; ; ; ; c = fraction{ 2 T }{ a sin beta } = fraction{ 2 * 12 }{ 3 sin(25° ) } = 18.93 ; ;

5. From area T, side a and side c we calculate side b - using Heron's formula for the area and solve of the bikvadratic equation:

s = fraction{ b+a+c }{ 2 } ; ; T**2 = s(s-a)(s-b)(s-c) ; ; ; ; s = fraction{ b+3+18.93 }{ 2 } = fraction{ b+21.93 }{ 2 } = b/2 + 10.965 ; ; ; ; T**2 = s(s-a)(s-b)(s-c) ; ; T**2 = ( b/2 + 10.965) ( b/2 + 10.965-b) ( b/2 + 10.965-3) ( b/2 + 10.965 - 18.93) ; ; ; ; 12**2 = ( b/2 + 10.965) ( 10.965-b/2) ( b/2 + 7.965) ( b/2 + (-7.965)) ; ; 2304 = ( b + 21.93) ( 21.93-b) ( b + 15.93) ( b + (-15.93)) ; ; ; ; D = a**2 * c**2 - 4 * S**2 = 3**2 * 18.93**2 - 4 * 12**2 = 2648.972 ; ; ; ; D_1 = -2 * sqrt{ D } + b**2 + c**2 = -2 * sqrt{ 2648.972 } + 3**2 + 18.93**2 = 264.394 ; ; D_2 = 2 * sqrt{ D } + b**2 + c**2 = 2 * sqrt{ 2648.972 } + 3**2 + 18.93**2 = 470.267 ; ; ; ; a_1 = sqrt{ D_1 } = sqrt{ 264.394 } = 16.26 ; ; a_2 = sqrt{ D_2 } = sqrt{ 470.267 } = 21.686 ; ;


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 = 16.26 ; ; c = 18.93 ; ;

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

p = a+b+c = 3+16.26+18.93 = 38.19 ; ;

7. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38.19 }{ 2 } = 19.09 ; ;

8. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.09 * (19.09-3)(19.09-16.26)(19.09-18.93) } ; ; T = sqrt{ 144 } = 12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12 }{ 3 } = 8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12 }{ 16.26 } = 1.48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12 }{ 18.93 } = 1.27 ; ;

10. 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{ 16.26**2+18.93**2-3**2 }{ 2 * 16.26 * 18.93 } ) = 4° 28'19" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3**2+18.93**2-16.26**2 }{ 2 * 3 * 18.93 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 4° 28'19" - 25° = 150° 31'41" ; ;

11. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12 }{ 19.09 } = 0.63 ; ;

12. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3 }{ 2 * sin 4° 28'19" } = 19.24 ; ;

13. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.26**2+2 * 18.93**2 - 3**2 } }{ 2 } = 17.582 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.93**2+2 * 3**2 - 16.26**2 } }{ 2 } = 10.843 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.26**2+2 * 3**2 - 18.93**2 } }{ 2 } = 6.864 ; ;
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