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.

Obtuse scalene triangle.

Sides: a = 24.55985797446   b = 19.55111142522   c = 43.5

Area: T = 78.95222653225
Perimeter: p = 87.61096939969
Semiperimeter: s = 43.80548469984

Angle ∠ A = α = 10.7° = 10°42' = 0.187675023 rad
Angle ∠ B = β = 8.5° = 8°30' = 0.14883529864 rad
Angle ∠ C = γ = 160.8° = 160°48' = 2.80664894372 rad

Height: ha = 6.43297093841
Height: hb = 8.07664977693
Height: hc = 3.63299892102

Median: ma = 31.40880733966
Median: mb = 33.94329727987
Median: mc = 4.43297238953

Inradius: r = 1.80223636819
Circumradius: R = 66.13662294282

Vertex coordinates: A[43.5; 0] B[0; 0] C[24.28988249491; 3.63299892102]
Centroid: CG[22.5966274983; 1.21099964034]
Coordinates of the circumscribed circle: U[21.75; -62.45774922886]
Coordinates of the inscribed circle: I[24.25437327462; 1.80223636819]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 169.3° = 169°18' = 0.187675023 rad
∠ B' = β' = 171.5° = 171°30' = 0.14883529864 rad
∠ C' = γ' = 19.2° = 19°12' = 2.80664894372 rad

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How did we calculate this triangle?

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

c = 43.5 ; ; alpha = 10.7° ; ; beta = 8.5° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 10.7 ° - 8.5 ° = 160.8 ° ; ;

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 = 43.5 * fraction{ sin 10° 42' }{ sin 160° 48' } = 24.56 ; ;

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{ 24.56**2+43.5**2 - 2 * 24.56 * 43.5 * cos 8° 30' } ; ; b = 19.55 ; ;


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 = 24.56 ; ; b = 19.55 ; ; c = 43.5 ; ;

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

p = a+b+c = 24.56+19.55+43.5 = 87.61 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 87.61 }{ 2 } = 43.8 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 43.8 * (43.8-24.56)(43.8-19.55)(43.8-43.5) } ; ; T = sqrt{ 6233.46 } = 78.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 78.95 }{ 24.56 } = 6.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 78.95 }{ 19.55 } = 8.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 78.95 }{ 43.5 } = 3.63 ; ;

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{ 19.55**2+43.5**2-24.56**2 }{ 2 * 19.55 * 43.5 } ) = 10° 42' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 24.56**2+43.5**2-19.55**2 }{ 2 * 24.56 * 43.5 } ) = 8° 30' ; ; gamma = 180° - alpha - beta = 180° - 10° 42' - 8° 30' = 160° 48' ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 78.95 }{ 43.8 } = 1.8 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 24.56 }{ 2 * sin 10° 42' } = 66.14 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.55**2+2 * 43.5**2 - 24.56**2 } }{ 2 } = 31.408 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 43.5**2+2 * 24.56**2 - 19.55**2 } }{ 2 } = 33.943 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.55**2+2 * 24.56**2 - 43.5**2 } }{ 2 } = 4.43 ; ;
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