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 = 3.27114899036   b = 0.68801809973   c = 3.2

Area: T = 1.08882895958
Perimeter: p = 7.15216709009
Semiperimeter: s = 3.57658354505

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 12° = 0.20994395102 rad
Angle ∠ C = γ = 78° = 1.36113568166 rad

Height: ha = 0.66553174106
Height: hb = 3.2
Height: hc = 0.68801809973

Median: ma = 1.63657449518
Median: mb = 3.21880213715
Median: mc = 1.73985759084

Inradius: r = 0.30443455469
Circumradius: R = 1.63657449518

Vertex coordinates: A[3.2; 0] B[0; 0] C[3.2; 0.68801809973]
Centroid: CG[2.13333333333; 0.22767269991]
Coordinates of the circumscribed circle: U[1.6; 0.34400904987]
Coordinates of the inscribed circle: I[2.89656544531; 0.30443455469]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 168° = 0.20994395102 rad
∠ C' = γ' = 102° = 1.36113568166 rad

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How did we calculate this triangle?

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

c = 3.2 ; ; alpha = 90° ; ; beta = 12° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 12 ° = 78 ° ; ;

3. From angle α, angle γ and side c we calculate 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 = 3.2 * fraction{ sin(90° ) }{ sin (78° ) } = 3.27 ; ;

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{ 3.27**2+3.2**2 - 2 * 3.27 * 3.2 * cos(12° ) } ; ; b = 0.68 ; ;


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 = 3.27 ; ; b = 0.68 ; ; c = 3.2 ; ;

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

p = a+b+c = 3.27+0.68+3.2 = 7.15 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 7.15 }{ 2 } = 3.58 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.58 * (3.58-3.27)(3.58-0.68)(3.58-3.2) } ; ; T = sqrt{ 1.18 } = 1.09 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.09 }{ 3.27 } = 0.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.09 }{ 0.68 } = 3.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.09 }{ 3.2 } = 0.68 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 3.27**2-0.68**2-3.2**2 }{ 2 * 0.68 * 3.2 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 0.68**2-3.27**2-3.2**2 }{ 2 * 3.27 * 3.2 } ) = 12° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.2**2-3.27**2-0.68**2 }{ 2 * 0.68 * 3.27 } ) = 78° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.09 }{ 3.58 } = 0.3 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.27 }{ 2 * sin 90° } = 1.64 ; ;




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