Triangle calculator

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 = 5.5   b = 5.82444584676   c = 10.32552432719

Area: T = 12
Perimeter: p = 21.65497017395
Semiperimeter: s = 10.82548508697

Angle ∠ A = α = 23.52204151622° = 23°31'14″ = 0.4110508686 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 131.4879584838° = 131°28'46″ = 2.29547516546 rad

Height: ha = 4.36436363636
Height: hb = 4.12105547492
Height: hc = 2.32444004396

Median: ma = 7.91986477717
Median: mb = 7.74326897912
Median: mc = 2.33112005628

Inradius: r = 1.10985603067
Circumradius: R = 6.89109214235

Vertex coordinates: A[10.32552432719; 0] B[0; 0] C[4.98546928287; 2.32444004396]
Centroid: CG[5.10333120335; 0.77548001465]
Coordinates of the circumscribed circle: U[5.1632621636; -4.56442234726]
Coordinates of the inscribed circle: I[55.0003924022; 1.10985603067]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.4879584838° = 156°28'46″ = 0.4110508686 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 48.52204151622° = 48°31'14″ = 2.29547516546 rad

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How did we calculate this triangle?

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

a = 5.5 ; ; beta = 25° ; ; S = 12 ; ;

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

h_c = a * sin beta = 5.5 * sin 25° = 2.324 ; ;

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 }{ 5.5 } = 4.364 ; ;

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 }{ 5.5 sin(25° ) } = 10.325 ; ;

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+5.5+10.325 }{ 2 } = fraction{ b+15.825 }{ 2 } = b/2 + 7.913 ; ; ; ; T**2 = s(s-a)(s-b)(s-c) ; ; T**2 = ( b/2 + 7.913) ( b/2 + 7.913-b) ( b/2 + 7.913-5.5) ( b/2 + 7.913 - 10.325) ; ; ; ; 12**2 = ( b/2 + 7.913) ( 7.913-b/2) ( b/2 + 2.413) ( b/2 + (-2.413)) ; ; 2304 = ( b + 15.825) ( 15.825-b) ( b + 4.825) ( b + (-4.825)) ; ; ; ; D = a**2 * c**2 - 4 * S**2 = 5.5**2 * 10.325**2 - 4 * 12**2 = 2648.972 ; ; ; ; D_1 = -2 * sqrt{ D } + b**2 + c**2 = -2 * sqrt{ 2648.972 } + 5.5**2 + 10.325**2 = 33.924 ; ; D_2 = 2 * sqrt{ D } + b**2 + c**2 = 2 * sqrt{ 2648.972 } + 5.5**2 + 10.325**2 = 239.797 ; ; ; ; a_1 = sqrt{ D_1 } = sqrt{ 33.924 } = 5.825 ; ; a_2 = sqrt{ D_2 } = sqrt{ 239.797 } = 15.485 ; ;


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 = 5.5 ; ; b = 5.82 ; ; c = 10.33 ; ;

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

p = a+b+c = 5.5+5.82+10.33 = 21.65 ; ;

7. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.65 }{ 2 } = 10.82 ; ;

8. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.82 * (10.82-5.5)(10.82-5.82)(10.82-10.33) } ; ; 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 }{ 5.5 } = 4.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12 }{ 5.82 } = 4.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12 }{ 10.33 } = 2.32 ; ;

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{ 5.82**2+10.33**2-5.5**2 }{ 2 * 5.82 * 10.33 } ) = 23° 31'14" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.5**2+10.33**2-5.82**2 }{ 2 * 5.5 * 10.33 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 23° 31'14" - 25° = 131° 28'46" ; ;

11. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12 }{ 10.82 } = 1.11 ; ;

12. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.5 }{ 2 * sin 23° 31'14" } = 6.89 ; ;

13. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.82**2+2 * 10.33**2 - 5.5**2 } }{ 2 } = 7.919 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.33**2+2 * 5.5**2 - 5.82**2 } }{ 2 } = 7.743 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.82**2+2 * 5.5**2 - 10.33**2 } }{ 2 } = 2.331 ; ;
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