Triangle calculator

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

You have entered side c, angle α and angle β.

Right scalene triangle.

Sides: a = 2.75884447974   b = 1.16657691454   c = 2.5

Area: T = 1.45772114317
Perimeter: p = 6.42442139428
Semiperimeter: s = 3.21221069714

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 65° = 1.13444640138 rad

Height: ha = 1.05765456544
Height: hb = 2.5
Height: hc = 1.16657691454

Median: ma = 1.37992223987
Median: mb = 2.5677051699
Median: mc = 1.70992447748

Inradius: r = 0.4543662174
Circumradius: R = 1.37992223987

Vertex coordinates: A[2.5; 0] B[0; 0] C[2.5; 1.16657691454]
Centroid: CG[1.66766666667; 0.38985897151]
Coordinates of the circumscribed circle: U[1.25; 0.58328845727]
Coordinates of the inscribed circle: I[2.0466337826; 0.4543662174]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 115° = 1.13444640138 rad

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How did we calculate this triangle?

1. Input data entered: side c, angle α and angle β.

c = 2.5 ; ; alpha = 90° ; ; beta = 25° ; ;

2. From angle α and angle β we calculate angle γ:

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 25 ° = 65 ° ; ;

3. From angle α, angle γ and side c we calculate side a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 2.5 * fraction{ sin 90° }{ sin 65° } = 2.76 ; ;

4. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 2.76**2+2.5**2 - 2 * 2.76 * 2.5 * cos 25° } ; ; b = 1.17 ; ;
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 = 2.76 ; ; b = 1.17 ; ; c = 2.5 ; ;

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

p = a+b+c = 2.76+1.17+2.5 = 6.42 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 6.42 }{ 2 } = 3.21 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.21 * (3.21-2.76)(3.21-1.17)(3.21-2.5) } ; ; T = sqrt{ 2.12 } = 1.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.46 }{ 2.76 } = 1.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.46 }{ 1.17 } = 2.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.46 }{ 2.5 } = 1.17 ; ;

9. 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{ 1.17**2+2.5**2-2.76**2 }{ 2 * 1.17 * 2.5 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.76**2+2.5**2-1.17**2 }{ 2 * 2.76 * 2.5 } ) = 25° ; ;
 gamma = 180° - alpha - beta = 180° - 90° - 25° = 65° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.46 }{ 3.21 } = 0.45 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 2.76 }{ 2 * sin 90° } = 1.38 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.17**2+2 * 2.5**2 - 2.76**2 } }{ 2 } = 1.379 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.5**2+2 * 2.76**2 - 1.17**2 } }{ 2 } = 2.567 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.17**2+2 * 2.76**2 - 2.5**2 } }{ 2 } = 1.709 ; ;
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