Triangle calculator

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

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

Obtuse scalene triangle.

Sides: a = 7.28   b = 100.3220225075   c = 98.06992100982

Area: T = 343.1433437795
Perimeter: p = 205.6699435173
Semiperimeter: s = 102.8354717587

Angle ∠ A = α = 4° = 0.07698131701 rad
Angle ∠ B = β = 106° = 1.85500490071 rad
Angle ∠ C = γ = 70° = 1.22217304764 rad

Height: ha = 94.27701752183
Height: hb = 6.84109622793
Height: hc = 6.99879851464

Median: ma = 99.13442986265
Median: mb = 48.15985640866
Median: mc = 51.51987391849

Inradius: r = 3.33768442667
Circumradius: R = 52.18215367754

Vertex coordinates: A[98.06992100982; 0] B[0; 0] C[-2.00766399503; 6.99879851464]
Centroid: CG[32.02108567159; 2.33326617155]
Coordinates of the circumscribed circle: U[49.03546050491; 17.84771366869]
Coordinates of the inscribed circle: I[2.51444925117; 3.33768442667]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 176° = 0.07698131701 rad
∠ B' = β' = 74° = 1.85500490071 rad
∠ C' = γ' = 110° = 1.22217304764 rad

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How did we calculate this triangle?

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

a = 7.28 ; ; beta = 106° ; ; gamma = 70° ; ;

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

 beta + gamma + alpha = 180° ; ; alpha = 180° - beta - gamma = 180° - 106 ° - 70 ° = 4 ° ; ;

3. From angle β, angle α and side a we calculate b - By using the law of sines, we calculate unknown side b:

 fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 7.28 * fraction{ sin(106° ) }{ sin (4° ) } = 100.32 ; ;

4. From angle γ, angle α and side a we calculate c - By using the law of sines, we calculate unknown side c:

 fraction{ c }{ a } = fraction{ sin( gamma ) }{ sin ( alpha ) } ; ; ; ; c = a * fraction{ sin( gamma ) }{ sin ( alpha ) } ; ; ; ; c = 7.28 * fraction{ sin(70° ) }{ sin (4° ) } = 98.07 ; ;


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 = 7.28 ; ; b = 100.32 ; ; c = 98.07 ; ;

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

p = a+b+c = 7.28+100.32+98.07 = 205.67 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 205.67 }{ 2 } = 102.83 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 102.83 * (102.83-7.28)(102.83-100.32)(102.83-98.07) } ; ; T = sqrt{ 117747.42 } = 343.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 343.14 }{ 7.28 } = 94.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 343.14 }{ 100.32 } = 6.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 343.14 }{ 98.07 } = 7 ; ;

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{ 7.28**2-100.32**2-98.07**2 }{ 2 * 100.32 * 98.07 } ) = 4° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 100.32**2-7.28**2-98.07**2 }{ 2 * 7.28 * 98.07 } ) = 106° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 98.07**2-7.28**2-100.32**2 }{ 2 * 100.32 * 7.28 } ) = 70° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 343.14 }{ 102.83 } = 3.34 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.28 }{ 2 * sin 4° } = 52.18 ; ;




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