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.

Acute scalene triangle.

Sides: a = 430.8660000999   b = 429.1277215462   c = 66

Area: T = 14143.06326736
Perimeter: p = 925.9877216461
Semiperimeter: s = 462.9943608231

Angle ∠ A = α = 87.1° = 87°6' = 1.52201817785 rad
Angle ∠ B = β = 84.1° = 84°6' = 1.46878219009 rad
Angle ∠ C = γ = 8.8° = 8°48' = 0.15435889742 rad

Height: ha = 65.65503859297
Height: hb = 65.91554775742
Height: hc = 428.5787656777

Median: ma = 218.7330424061
Median: mb = 221.2770487114
Median: mc = 428.7266315679

Inradius: r = 30.54769933541
Circumradius: R = 215.706624315

Vertex coordinates: A[66; 0] B[0; 0] C[44.28991925028; 428.5787656777]
Centroid: CG[36.76330641676; 142.8599218926]
Coordinates of the circumscribed circle: U[33; 213.1677031536]
Coordinates of the inscribed circle: I[33.86663927683; 30.54769933541]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 92.9° = 92°54' = 1.52201817785 rad
∠ B' = β' = 95.9° = 95°54' = 1.46878219009 rad
∠ C' = γ' = 171.2° = 171°12' = 0.15435889742 rad

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How did we calculate this triangle?

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

c = 66 ; ; alpha = 87.1° ; ; beta = 84.1° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 87.1 ° - 84.1 ° = 8.8 ° ; ;

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 = 66 * fraction{ sin 87° 6' }{ sin 8° 48' } = 430.86 ; ;

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{ 430.86**2+66**2 - 2 * 430.86 * 66 * cos 84° 6' } ; ; b = 429.13 ; ;


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 = 430.86 ; ; b = 429.13 ; ; c = 66 ; ;

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

p = a+b+c = 430.86+429.13+66 = 925.99 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 925.99 }{ 2 } = 462.99 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 462.99 * (462.99-430.86)(462.99-429.13)(462.99-66) } ; ; T = sqrt{ 200026221.79 } = 14143.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14143.06 }{ 430.86 } = 65.65 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14143.06 }{ 429.13 } = 65.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14143.06 }{ 66 } = 428.58 ; ;

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{ 429.13**2+66**2-430.86**2 }{ 2 * 429.13 * 66 } ) = 87° 6' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 430.86**2+66**2-429.13**2 }{ 2 * 430.86 * 66 } ) = 84° 6' ; ; gamma = 180° - alpha - beta = 180° - 87° 6' - 84° 6' = 8° 48' ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14143.06 }{ 462.99 } = 30.55 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 430.86 }{ 2 * sin 87° 6' } = 215.71 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 429.13**2+2 * 66**2 - 430.86**2 } }{ 2 } = 218.73 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 66**2+2 * 430.86**2 - 429.13**2 } }{ 2 } = 221.27 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 429.13**2+2 * 430.86**2 - 66**2 } }{ 2 } = 428.726 ; ;
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