Triangle calculator

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

You have entered side b, angle α and angle β.

Right scalene triangle.

Sides: a = 128.1155427326   b = 495   c = 478.1333284013

Area: T = 30628.125
Perimeter: p = 1101.249871134
Semiperimeter: s = 550.6244355669

Angle ∠ A = α = 15° = 0.26217993878 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 75° = 1.3098996939 rad

Height: ha = 478.1333284013
Height: hb = 123.75
Height: hc = 128.1155427326

Median: ma = 482.4055252833
Median: mb = 247.5
Median: mc = 271.2311307262

Inradius: r = 55.62443556694
Circumradius: R = 247.5

Vertex coordinates: A[478.1333284013; 0] B[0; 0] C[-0; 128.1155427326]
Centroid: CG[159.3787761338; 42.70551424419]
Coordinates of the circumscribed circle: U[239.0676642006; 64.05877136629]
Coordinates of the inscribed circle: I[55.62443556694; 55.62443556694]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165° = 0.26217993878 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 105° = 1.3098996939 rad

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How did we calculate this triangle?

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

b = 495 ; ; alpha = 15° ; ; beta = 90° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 15 ° - 90 ° = 75 ° ; ;

3. From angle α, angle β and side b we calculate side a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ b } = fraction{ sin alpha }{ sin beta } ; ; ; ; a = b * fraction{ sin alpha }{ sin beta } ; ; ; ; a = 495 * fraction{ sin 15° }{ sin 90° } = 128.12 ; ;

4. Calculation of the third side c of the triangle using a Law of Cosines

c**2 = b**2+a**2 - 2ba cos gamma ; ; c = sqrt{ b**2+a**2 - 2ba cos gamma } ; ; c = sqrt{ 495**2+128.12**2 - 2 * 495 * 128.12 * cos 75° } ; ; c = 478.13 ; ;
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 = 128.12 ; ; b = 495 ; ; c = 478.13 ; ;

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

p = a+b+c = 128.12+495+478.13 = 1101.25 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1101.25 }{ 2 } = 550.62 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 550.62 * (550.62-128.12)(550.62-495)(550.62-478.13) } ; ; T = sqrt{ 938082041.02 } = 30628.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 30628.13 }{ 128.12 } = 478.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 30628.13 }{ 495 } = 123.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 30628.13 }{ 478.13 } = 128.12 ; ;

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{ 495**2+478.13**2-128.12**2 }{ 2 * 495 * 478.13 } ) = 15° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 128.12**2+478.13**2-495**2 }{ 2 * 128.12 * 478.13 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 15° - 90° = 75° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 30628.13 }{ 550.62 } = 55.62 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 128.12 }{ 2 * sin 15° } = 247.5 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 495**2+2 * 478.13**2 - 128.12**2 } }{ 2 } = 482.405 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 478.13**2+2 * 128.12**2 - 495**2 } }{ 2 } = 247.5 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 495**2+2 * 128.12**2 - 478.13**2 } }{ 2 } = 271.231 ; ;
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