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.

Acute scalene triangle.

Sides: a = 481.8222115079   b = 383.4122350648   c = 350

Area: T = 66444.17664013
Perimeter: p = 1215.234446573
Semiperimeter: s = 607.6177232863

Angle ∠ A = α = 82° = 1.43111699866 rad
Angle ∠ B = β = 52° = 0.9087571211 rad
Angle ∠ C = γ = 46° = 0.80328514559 rad

Height: ha = 275.8043763762
Height: hb = 346.594382406
Height: hc = 379.6811008008

Median: ma = 276.9743604645
Median: mb = 374.933335092
Median: mc = 398.6990093436

Inradius: r = 109.3522027572
Circumradius: R = 243.2798628428

Vertex coordinates: A[350; 0] B[0; 0] C[296.6399314214; 379.6811008008]
Centroid: CG[215.5466438071; 126.5660336003]
Coordinates of the circumscribed circle: U[175; 168.9965535591]
Coordinates of the inscribed circle: I[224.2054882215; 109.3522027572]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 98° = 1.43111699866 rad
∠ B' = β' = 128° = 0.9087571211 rad
∠ C' = γ' = 134° = 0.80328514559 rad

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How did we calculate this triangle?

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

c = 350 ; ; alpha = 82° ; ; beta = 52° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 82 ° - 52 ° = 46 ° ; ;

3. From angle α, angle γ and side c we calculate side 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 = 350 * fraction{ sin 82° }{ sin 46° } = 481.82 ; ;

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{ 481.82**2+350**2 - 2 * 481.82 * 350 * cos(52° ) } ; ; b = 383.41 ; ;


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 = 481.82 ; ; b = 383.41 ; ; c = 350 ; ;

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

p = a+b+c = 481.82+383.41+350 = 1215.23 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1215.23 }{ 2 } = 607.62 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 607.62 * (607.62-481.82)(607.62-383.41)(607.62-350) } ; ; T = sqrt{ 4414828577.65 } = 66444.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 66444.18 }{ 481.82 } = 275.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 66444.18 }{ 383.41 } = 346.59 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 66444.18 }{ 350 } = 379.68 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 383.41**2+350**2-481.82**2 }{ 2 * 383.41 * 350 } ) = 82° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 481.82**2+350**2-383.41**2 }{ 2 * 481.82 * 350 } ) = 52° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 481.82**2+383.41**2-350**2 }{ 2 * 481.82 * 383.41 } ) = 46° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 66444.18 }{ 607.62 } = 109.35 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 481.82 }{ 2 * sin 82° } = 243.28 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 383.41**2+2 * 350**2 - 481.82**2 } }{ 2 } = 276.974 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 350**2+2 * 481.82**2 - 383.41**2 } }{ 2 } = 374.933 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 383.41**2+2 * 481.82**2 - 350**2 } }{ 2 } = 398.69 ; ;
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