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 = 1.04443258315   b = 0.8   c = 0.67112797049

Area: T = 0.2698511882
Perimeter: p = 2.51656055364
Semiperimeter: s = 1.25878027682

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 40° = 0.69881317008 rad

Height: ha = 0.51442300877
Height: hb = 0.67112797049
Height: hc = 0.8

Median: ma = 0.52221629157
Median: mb = 0.78114195047
Median: mc = 0.8687556402

Inradius: r = 0.21334769367
Circumradius: R = 0.52221629157

Vertex coordinates: A[0.67112797049; 0] B[0; 0] C[0.67112797049; 0.8]
Centroid: CG[0.44875198033; 0.26766666667]
Coordinates of the circumscribed circle: U[0.33656398525; 0.4]
Coordinates of the inscribed circle: I[0.45878027682; 0.21334769367]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 140° = 0.69881317008 rad

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How did we calculate this triangle?

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

b = 0.8 ; ; alpha = 90° ; ; beta = 50° ; ;

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

alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 50 ° = 40 ° ; ;

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 = 0.8 * fraction{ sin 90° }{ sin 50° } = 1.04 ; ;

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{ 0.8**2+1.04**2 - 2 * 0.8 * 1.04 * cos 40° } ; ; c = 0.67 ; ;


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 = 1.04 ; ; b = 0.8 ; ; c = 0.67 ; ;

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

p = a+b+c = 1.04+0.8+0.67 = 2.52 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2.52 }{ 2 } = 1.26 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1.26 * (1.26-1.04)(1.26-0.8)(1.26-0.67) } ; ; T = sqrt{ 0.07 } = 0.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.27 }{ 1.04 } = 0.51 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.27 }{ 0.8 } = 0.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.27 }{ 0.67 } = 0.8 ; ;

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{ 0.8**2+0.67**2-1.04**2 }{ 2 * 0.8 * 0.67 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.04**2+0.67**2-0.8**2 }{ 2 * 1.04 * 0.67 } ) = 50° ; ; gamma = 180° - alpha - beta = 180° - 90° - 50° = 40° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.27 }{ 1.26 } = 0.21 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.04 }{ 2 * sin 90° } = 0.52 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.8**2+2 * 0.67**2 - 1.04**2 } }{ 2 } = 0.522 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.67**2+2 * 1.04**2 - 0.8**2 } }{ 2 } = 0.781 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.8**2+2 * 1.04**2 - 0.67**2 } }{ 2 } = 0.868 ; ;
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