Triangle calculator

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

You have entered side b, height hc, angle α and angle γ.

Obtuse scalene triangle.

Sides: a = 49.99331552466   b = 111.8   c = 99.99326420241

Area: T = 2499.474381375
Perimeter: p = 261.7865797271
Semiperimeter: s = 130.8932898635

Angle ∠ A = α = 26.562° = 26°33'43″ = 0.46435943559 rad
Angle ∠ B = β = 90.008° = 90°29″ = 1.57109359531 rad
Angle ∠ C = γ = 63.43° = 63°25'48″ = 1.10770623445 rad

Height: ha = 99.99326410494
Height: hb = 44.71333061494
Height: hc = 49.99331547593

Median: ma = 103.0733058248
Median: mb = 55.8943756496
Median: mc = 70.708817259

Inradius: r = 19.0965564693
Circumradius: R = 55.99000005449

Vertex coordinates: A[99.99326420241; 0] B[0; 0] C[-0.00769803613; 49.99331547593]
Centroid: CG[33.32985538876; 16.66443849198]
Coordinates of the circumscribed circle: U[49.9966321012; 25.00435586703]
Coordinates of the inscribed circle: I[19.09328986353; 19.0965564693]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.438° = 153°26'17″ = 0.46435943559 rad
∠ B' = β' = 89.992° = 89°59'31″ = 1.57109359531 rad
∠ C' = γ' = 116.57° = 116°34'12″ = 1.10770623445 rad

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How did we calculate this triangle?

1. Input data entered: side b, angle α, angle γ and height hc.

b = 111.8 ; ; alpha = 26.562° ; ; gamma = 63.43° ; ; hc = 50 ; ;

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

 alpha + gamma + beta = 180° ; ; beta = 180° - alpha - gamma = 180° - 26.562 ° - 63.43 ° = 90.008 ° ; ;

3. From angle α, angle β and side b we calculate 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 = 111.8 * fraction{ sin(26° 33'43") }{ sin (90° 29") } = 49.99 ; ;

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{ 111.8**2+49.99**2 - 2 * 111.8 * 49.99 * cos(63° 25'48") } ; ; c = 99.99 ; ;


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 = 49.99 ; ; b = 111.8 ; ; c = 99.99 ; ;

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

p = a+b+c = 49.99+111.8+99.99 = 261.79 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 261.79 }{ 2 } = 130.89 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 130.89 * (130.89-49.99)(130.89-111.8)(130.89-99.99) } ; ; T = sqrt{ 6247369.35 } = 2499.47 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2499.47 }{ 49.99 } = 99.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2499.47 }{ 111.8 } = 44.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2499.47 }{ 99.99 } = 49.99 ; ;

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{ 49.99**2-111.8**2-99.99**2 }{ 2 * 111.8 * 99.99 } ) = 26° 33'43" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 111.8**2-49.99**2-99.99**2 }{ 2 * 49.99 * 99.99 } ) = 90° 29" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 99.99**2-49.99**2-111.8**2 }{ 2 * 111.8 * 49.99 } ) = 63° 25'48" ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2499.47 }{ 130.89 } = 19.1 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 49.99 }{ 2 * sin 26° 33'43" } = 55.9 ; ;




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