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 = 100.3821983754   b = 8.74988663526   c = 100

Area: T = 437.443331763
Perimeter: p = 209.1310850107
Semiperimeter: s = 104.5655425053

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 5° = 0.08772664626 rad
Angle ∠ C = γ = 85° = 1.48435298642 rad

Height: ha = 8.71655742748
Height: hb = 100
Height: hc = 8.74988663526

Median: ma = 50.19109918772
Median: mb = 100.09656326
Median: mc = 50.76596558544

Inradius: r = 4.18334412991
Circumradius: R = 50.19109918772

Vertex coordinates: A[100; 0] B[0; 0] C[100; 8.74988663526]
Centroid: CG[66.66766666667; 2.91662887842]
Coordinates of the circumscribed circle: U[50; 4.37444331763]
Coordinates of the inscribed circle: I[95.81765587009; 4.18334412991]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 175° = 0.08772664626 rad
∠ C' = γ' = 95° = 1.48435298642 rad

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How did we calculate this triangle?

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

c = 100 ; ; alpha = 90° ; ; beta = 5° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 5 ° = 85 ° ; ;

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 = 100 * fraction{ sin 90° }{ sin 85° } = 100.38 ; ;

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{ 100.38**2+100**2 - 2 * 100.38 * 100 * cos 5° } ; ; b = 8.75 ; ;
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 = 100.38 ; ; b = 8.75 ; ; c = 100 ; ;

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

p = a+b+c = 100.38+8.75+100 = 209.13 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 209.13 }{ 2 } = 104.57 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 104.57 * (104.57-100.38)(104.57-8.75)(104.57-100) } ; ; T = sqrt{ 191356.66 } = 437.44 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 437.44 }{ 100.38 } = 8.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 437.44 }{ 8.75 } = 100 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 437.44 }{ 100 } = 8.75 ; ;

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{ 8.75**2+100**2-100.38**2 }{ 2 * 8.75 * 100 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100.38**2+100**2-8.75**2 }{ 2 * 100.38 * 100 } ) = 5° ; ;
 gamma = 180° - alpha - beta = 180° - 90° - 5° = 85° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 437.44 }{ 104.57 } = 4.18 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100.38 }{ 2 * sin 90° } = 50.19 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.75**2+2 * 100**2 - 100.38**2 } }{ 2 } = 50.191 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 100.38**2 - 8.75**2 } }{ 2 } = 100.096 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.75**2+2 * 100.38**2 - 100**2 } }{ 2 } = 50.76 ; ;
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