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 = 1003.461090197   b = 83.26993327934   c = 1000

Area: T = 41634.66663967
Perimeter: p = 2086.733023476
Semiperimeter: s = 1043.365511738

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 4.76° = 4°45'36″ = 0.08330776724 rad
Angle ∠ C = γ = 85.24° = 85°14'24″ = 1.48877186544 rad

Height: ha = 82.98221397424
Height: hb = 1000
Height: hc = 83.26993327934

Median: ma = 501.7330450985
Median: mb = 1000.866634744
Median: mc = 506.8866359832

Inradius: r = 39.90442154113
Circumradius: R = 501.7330450985

Vertex coordinates: A[1000; 0] B[0; 0] C[1000; 83.26993327934]
Centroid: CG[666.6676666667; 27.75664442645]
Coordinates of the circumscribed circle: U[500; 41.63546663967]
Coordinates of the inscribed circle: I[960.0965784589; 39.90442154113]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 175.24° = 175°14'24″ = 0.08330776724 rad
∠ C' = γ' = 94.76° = 94°45'36″ = 1.48877186544 rad

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How did we calculate this triangle?

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

c = 1000 ; ; alpha = 90° ; ; beta = 4.76° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 4.76 ° = 85.24 ° ; ;

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 = 1000 * fraction{ sin 90° }{ sin 85° 14'24" } = 1003.46 ; ;

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{ 1003.46**2+1000**2 - 2 * 1003.46 * 1000 * cos 4° 45'36" } ; ; b = 83.27 ; ;
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 = 1003.46 ; ; b = 83.27 ; ; c = 1000 ; ;

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

p = a+b+c = 1003.46+83.27+1000 = 2086.73 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2086.73 }{ 2 } = 1043.37 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1043.37 * (1043.37-1003.46)(1043.37-83.27)(1043.37-1000) } ; ; T = sqrt{ 1733445445.96 } = 41634.67 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 41634.67 }{ 1003.46 } = 82.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 41634.67 }{ 83.27 } = 1000 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 41634.67 }{ 1000 } = 83.27 ; ;

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{ 83.27**2+1000**2-1003.46**2 }{ 2 * 83.27 * 1000 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1003.46**2+1000**2-83.27**2 }{ 2 * 1003.46 * 1000 } ) = 4° 45'36" ; ;
 gamma = 180° - alpha - beta = 180° - 90° - 4° 45'36" = 85° 14'24" ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 41634.67 }{ 1043.37 } = 39.9 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1003.46 }{ 2 * sin 90° } = 501.73 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 83.27**2+2 * 1000**2 - 1003.46**2 } }{ 2 } = 501.73 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1000**2+2 * 1003.46**2 - 83.27**2 } }{ 2 } = 1000.866 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 83.27**2+2 * 1003.46**2 - 1000**2 } }{ 2 } = 506.886 ; ;
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