Triangle calculator

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

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

Right scalene triangle.

Sides: a = 17.45224064373   b = 999.8487695156   c = 1000

Area: T = 8724.874417563
Perimeter: p = 2017.330010159
Semiperimeter: s = 1008.65500508

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

Height: ha = 999.8487695156
Height: hb = 17.45224064373
Height: hc = 17.45497483513

Median: ma = 999.8865773542
Median: mb = 500.2288387707
Median: mc = 500

Inradius: r = 8.65500507968
Circumradius: R = 500

Vertex coordinates: A[1000; 0] B[0; 0] C[0.30545864905; 17.45497483513]
Centroid: CG[333.4354862163; 5.81765827838]
Coordinates of the circumscribed circle: U[500; 0]
Coordinates of the inscribed circle: I[8.80223556404; 8.65500507968]

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

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How did we calculate this triangle?

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

c = 1000 ; ; alpha = 1° ; ; gamma = 90° ; ;

2. 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 1° }{ sin 90° } = 17.45 ; ;

3. 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{ 17.45**2+1000**2 - 2 * 17.45 * 1000 * cos 89° } ; ; b = 999.85 ; ;
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 = 17.45 ; ; b = 999.85 ; ; c = 1000 ; ;

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

p = a+b+c = 17.45+999.85+1000 = 2017.3 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2017.3 }{ 2 } = 1008.65 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1008.65 * (1008.65-17.45)(1008.65-999.85)(1008.65-1000) } ; ; T = sqrt{ 76123429.38 } = 8724.87 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8724.87 }{ 17.45 } = 999.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8724.87 }{ 999.85 } = 17.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8724.87 }{ 1000 } = 17.45 ; ;

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{ 999.85**2+1000**2-17.45**2 }{ 2 * 999.85 * 1000 } ) = 1° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 17.45**2+1000**2-999.85**2 }{ 2 * 17.45 * 1000 } ) = 89° ; ; gamma = 180° - alpha - beta = 180° - 1° - 89° = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8724.87 }{ 1008.65 } = 8.65 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 17.45 }{ 2 * sin 1° } = 500 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 999.85**2+2 * 1000**2 - 17.45**2 } }{ 2 } = 999.886 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1000**2+2 * 17.45**2 - 999.85**2 } }{ 2 } = 500.228 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 999.85**2+2 * 17.45**2 - 1000**2 } }{ 2 } = 500 ; ;
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