Triangle calculator

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

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

Right scalene triangle.

Sides: a = 0.35443872993   b = 3.21   c = 3.23295031132

Area: T = 0.56987916154
Perimeter: p = 6.79438904124
Semiperimeter: s = 3.39769452062

Angle ∠ A = α = 6.3° = 6°18' = 0.11099557429 rad
Angle ∠ B = β = 83.7° = 83°42' = 1.46108405839 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 3.21
Height: hb = 0.35443872993
Height: hc = 0.35222471386

Median: ma = 3.21548868704
Median: mb = 1.64436591368
Median: mc = 1.61547515566

Inradius: r = 0.16774420931
Circumradius: R = 1.61547515566

Vertex coordinates: A[3.23295031132; 0] B[0; 0] C[0.03988884461; 0.35222471386]
Centroid: CG[1.08994638531; 0.11774157129]
Coordinates of the circumscribed circle: U[1.61547515566; -0]
Coordinates of the inscribed circle: I[0.18769452062; 0.16774420931]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 173.7° = 173°42' = 0.11099557429 rad
∠ B' = β' = 96.3° = 96°18' = 1.46108405839 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

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

b = 3.21 ; ; beta = 83.7° ; ; gamma = 90° ; ;

2. From angle γ, angle β and side b we calculate side c - By using the law of sines, we calculate unknown side c:

 fraction{ c }{ b } = fraction{ sin gamma }{ sin beta } ; ; ; ; c = b * fraction{ sin gamma }{ sin beta } ; ; ; ; c = 3.21 * fraction{ sin 90° }{ sin 83° 42' } = 3.23 ; ;

3. Calculation of the third side a of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 3.21**2+3.23**2 - 2 * 3.21 * 3.23 * cos 6° 18' } ; ; a = 0.35 ; ;
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 = 0.35 ; ; b = 3.21 ; ; c = 3.23 ; ;

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

p = a+b+c = 0.35+3.21+3.23 = 6.79 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 6.79 }{ 2 } = 3.4 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.4 * (3.4-0.35)(3.4-3.21)(3.4-3.23) } ; ; T = sqrt{ 0.32 } = 0.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.57 }{ 0.35 } = 3.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.57 }{ 3.21 } = 0.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.57 }{ 3.23 } = 0.35 ; ;

8. 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{ 3.21**2+3.23**2-0.35**2 }{ 2 * 3.21 * 3.23 } ) = 6° 18' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 0.35**2+3.23**2-3.21**2 }{ 2 * 0.35 * 3.23 } ) = 83° 42' ; ; gamma = 180° - alpha - beta = 180° - 6° 18' - 83° 42' = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.57 }{ 3.4 } = 0.17 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 0.35 }{ 2 * sin 6° 18' } = 1.61 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.21**2+2 * 3.23**2 - 0.35**2 } }{ 2 } = 3.215 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.23**2+2 * 0.35**2 - 3.21**2 } }{ 2 } = 1.644 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.21**2+2 * 0.35**2 - 3.23**2 } }{ 2 } = 1.615 ; ;
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