Triangle calculator

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

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

Right scalene triangle.

Sides: a = 76.78875858022   b = 133   c = 153.5755171604

Area: T = 5106.374445585
Perimeter: p = 363.3632757407
Semiperimeter: s = 181.6811378703

Angle ∠ A = α = 30° = 0.52435987756 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 133
Height: hb = 76.78875858022
Height: hc = 66.5

Median: ma = 138.4310788965
Median: mb = 101.5880427905
Median: mc = 76.78875858022

Inradius: r = 28.10662070989
Circumradius: R = 76.78875858022

Vertex coordinates: A[153.5755171604; 0] B[0; 0] C[38.39437929011; 66.5]
Centroid: CG[63.99896548352; 22.16766666667]
Coordinates of the circumscribed circle: U[76.78875858022; 0]
Coordinates of the inscribed circle: I[48.68113787033; 28.10662070989]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150° = 0.52435987756 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

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

b = 133 ; ; alpha = 30° ; ; gamma = 90° ; ;

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

 alpha + gamma + beta = 180° ; ; beta = 180° - alpha - gamma = 180° - 30 ° - 90 ° = 60 ° ; ;

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 = 133 * fraction{ sin 30° }{ sin 60° } = 76.79 ; ;

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{ 133**2+76.79**2 - 2 * 133 * 76.79 * cos 90° } ; ; c = 153.58 ; ;
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 = 76.79 ; ; b = 133 ; ; c = 153.58 ; ;

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

p = a+b+c = 76.79+133+153.58 = 363.36 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 363.36 }{ 2 } = 181.68 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 181.68 * (181.68-76.79)(181.68-133)(181.68-153.58) } ; ; T = sqrt{ 26075060.08 } = 5106.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5106.37 }{ 76.79 } = 133 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5106.37 }{ 133 } = 76.79 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5106.37 }{ 153.58 } = 66.5 ; ;

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{ 133**2+153.58**2-76.79**2 }{ 2 * 133 * 153.58 } ) = 30° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 76.79**2+153.58**2-133**2 }{ 2 * 76.79 * 153.58 } ) = 60° ; ; gamma = 180° - alpha - beta = 180° - 30° - 60° = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5106.37 }{ 181.68 } = 28.11 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 76.79 }{ 2 * sin 30° } = 76.79 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 133**2+2 * 153.58**2 - 76.79**2 } }{ 2 } = 138.431 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 153.58**2+2 * 76.79**2 - 133**2 } }{ 2 } = 101.58 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 133**2+2 * 76.79**2 - 153.58**2 } }{ 2 } = 76.788 ; ;
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