Triangle calculator

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

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

Right scalene triangle.

Sides: a = 3.03442892771   b = 1.28223460599   c = 2.75

Area: T = 1.76332258324
Perimeter: p = 7.06766353371
Semiperimeter: s = 3.53333176685

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 65° = 1.13444640138 rad

Height: ha = 1.16222002198
Height: hb = 2.75
Height: hc = 1.28223460599

Median: ma = 1.51771446386
Median: mb = 2.82437568688
Median: mc = 1.88801692523

Inradius: r = 0.49990283914
Circumradius: R = 1.51771446386

Vertex coordinates: A[2.75; 0] B[0; 0] C[2.75; 1.28223460599]
Centroid: CG[1.83333333333; 0.42774486866]
Coordinates of the circumscribed circle: U[1.375; 0.641117303]
Coordinates of the inscribed circle: I[2.25109716086; 0.49990283914]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 115° = 1.13444640138 rad

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How did we calculate this triangle?

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

c = 2.75 ; ; beta = 25° ; ; gamma = 65° ; ;

2. From angle β, angle γ and side c we calculate side b - By using the law of sines, we calculate unknown side b:

 fraction{ b }{ c } = fraction{ sin beta }{ sin gamma } ; ; ; ; b = c * fraction{ sin beta }{ sin gamma } ; ; ; ; b = 2.75 * fraction{ sin 25° }{ sin 65° } = 1.28 ; ;

3. Calculation of the third side a of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 1.28**2+2.75**2 - 2 * 1.28 * 2.75 * cos 90° } ; ; a = 3.03 ; ;


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 = 3.03 ; ; b = 1.28 ; ; c = 2.75 ; ;

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

p = a+b+c = 3.03+1.28+2.75 = 7.07 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 7.07 }{ 2 } = 3.53 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.53 * (3.53-3.03)(3.53-1.28)(3.53-2.75) } ; ; T = sqrt{ 3.11 } = 1.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.76 }{ 3.03 } = 1.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.76 }{ 1.28 } = 2.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.76 }{ 2.75 } = 1.28 ; ;

8. 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{ 1.28**2+2.75**2-3.03**2 }{ 2 * 1.28 * 2.75 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.03**2+2.75**2-1.28**2 }{ 2 * 3.03 * 2.75 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 90° - 25° = 65° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.76 }{ 3.53 } = 0.5 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.03 }{ 2 * sin 90° } = 1.52 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.28**2+2 * 2.75**2 - 3.03**2 } }{ 2 } = 1.517 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.75**2+2 * 3.03**2 - 1.28**2 } }{ 2 } = 2.824 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.28**2+2 * 3.03**2 - 2.75**2 } }{ 2 } = 1.88 ; ;
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