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.

Right scalene triangle.

Sides: a = 1000   b = 17.45550649282   c = 1000.152232804

Area: T = 8727.532246411
Perimeter: p = 2017.607739297
Semiperimeter: s = 1008.804369649

Angle ∠ A = α = 89° = 1.55333430343 rad
Angle ∠ B = β = 1° = 0.01774532925 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 17.45550649282
Height: hb = 1000
Height: hc = 17.45224064373

Median: ma = 500.3054586519
Median: mb = 1000.038808419
Median: mc = 500.0766164022

Inradius: r = 8.65113684422
Circumradius: R = 500.0766164022

Vertex coordinates: A[1000.152232804; 0] B[0; 0] C[999.8487695156; 17.45224064373]
Centroid: CG[666.66766744; 5.81774688124]
Coordinates of the circumscribed circle: U[500.0766164022; -0]
Coordinates of the inscribed circle: I[991.3498631558; 8.65113684422]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 91° = 1.55333430343 rad
∠ B' = β' = 179° = 0.01774532925 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

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

a = 1000 ; ; beta = 1° ; ; gamma = 90° ; ;

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

 beta + gamma + alpha = 180° ; ; alpha = 180° - beta - gamma = 180° - 1 ° - 90 ° = 89 ° ; ;

3. From angle β, angle α and side a we calculate side 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 = 1000 * fraction{ sin 1° }{ sin 89° } = 17.46 ; ;

4. From angle γ, angle α and side a we calculate side 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 = 1000 * fraction{ sin 90° }{ sin 89° } = 1000.15 ; ;


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 = 1000 ; ; b = 17.46 ; ; c = 1000.15 ; ;

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

p = a+b+c = 1000+17.46+1000.15 = 2017.61 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2017.61 }{ 2 } = 1008.8 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1008.8 * (1008.8-1000)(1008.8-17.46)(1008.8-1000.15) } ; ; T = sqrt{ 76169822.91 } = 8727.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8727.53 }{ 1000 } = 17.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8727.53 }{ 17.46 } = 1000 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8727.53 }{ 1000.15 } = 17.45 ; ;

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{ 17.46**2+1000.15**2-1000**2 }{ 2 * 17.46 * 1000.15 } ) = 89° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1000**2+1000.15**2-17.46**2 }{ 2 * 1000 * 1000.15 } ) = 1° ; ; gamma = 180° - alpha - beta = 180° - 89° - 1° = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8727.53 }{ 1008.8 } = 8.65 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1000 }{ 2 * sin 89° } = 500.08 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.46**2+2 * 1000.15**2 - 1000**2 } }{ 2 } = 500.305 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1000.15**2+2 * 1000**2 - 17.46**2 } }{ 2 } = 1000.038 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.46**2+2 * 1000**2 - 1000.15**2 } }{ 2 } = 500.076 ; ;
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