Triangle calculator

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

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

Right scalene triangle.

Sides: a = 398.7943633869   b = 240   c = 318.4910757189

Area: T = 38218.89108627
Perimeter: p = 957.2844391058
Semiperimeter: s = 478.6422195529

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 37° = 0.64657718232 rad
Angle ∠ C = γ = 53° = 0.92550245036 rad

Height: ha = 191.6732522411
Height: hb = 318.4910757189
Height: hc = 240

Median: ma = 199.3976816935
Median: mb = 340.3477414291
Median: mc = 288.0266197773

Inradius: r = 79.84985616598
Circumradius: R = 199.3976816935

Vertex coordinates: A[318.4910757189; 0] B[0; 0] C[318.4910757189; 240]
Centroid: CG[212.3277171459; 80]
Coordinates of the circumscribed circle: U[159.2455378594; 120]
Coordinates of the inscribed circle: I[238.6422195529; 79.84985616598]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 143° = 0.64657718232 rad
∠ C' = γ' = 127° = 0.92550245036 rad

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How did we calculate this triangle?

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

b = 240 ; ; alpha = 90° ; ; beta = 37° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 37 ° = 53 ° ; ;

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

 fraction{ a }{ b } = fraction{ sin alpha }{ sin beta } ; ; ; ; a = b * fraction{ sin alpha }{ sin beta } ; ; ; ; a = 240 * fraction{ sin 90° }{ sin 37° } = 398.79 ; ;

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

c**2 = b**2+a**2 - 2ba cos gamma ; ; c = sqrt{ b**2+a**2 - 2ba cos gamma } ; ; c = sqrt{ 240**2+398.79**2 - 2 * 240 * 398.79 * cos 53° } ; ; c = 318.49 ; ;


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 = 398.79 ; ; b = 240 ; ; c = 318.49 ; ;

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

p = a+b+c = 398.79+240+318.49 = 957.28 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 957.28 }{ 2 } = 478.64 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 478.64 * (478.64-398.79)(478.64-240)(478.64-318.49) } ; ; T = sqrt{ 1460683618.77 } = 38218.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 38218.89 }{ 398.79 } = 191.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 38218.89 }{ 240 } = 318.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 38218.89 }{ 318.49 } = 240 ; ;

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{ 240**2+318.49**2-398.79**2 }{ 2 * 240 * 318.49 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 398.79**2+318.49**2-240**2 }{ 2 * 398.79 * 318.49 } ) = 37° ; ; gamma = 180° - alpha - beta = 180° - 90° - 37° = 53° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 38218.89 }{ 478.64 } = 79.85 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 398.79 }{ 2 * sin 90° } = 199.4 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 240**2+2 * 318.49**2 - 398.79**2 } }{ 2 } = 199.397 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 318.49**2+2 * 398.79**2 - 240**2 } }{ 2 } = 340.347 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 240**2+2 * 398.79**2 - 318.49**2 } }{ 2 } = 288.026 ; ;
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