Triangle calculator SAS

Please enter two sides of the triangle and the included angle
°


Acute scalene triangle.

Sides: a = 26.7   b = 31.2   c = 27.03215698938

Area: T = 341.1933209487
Perimeter: p = 84.93215698938
Semiperimeter: s = 42.46657849469

Angle ∠ A = α = 54.00985117211° = 54°31″ = 0.94326263536 rad
Angle ∠ B = β = 70.99114882789° = 70°59'29″ = 1.23990352114 rad
Angle ∠ C = γ = 55° = 0.96599310886 rad

Height: ha = 25.55875437818
Height: hb = 21.87113595825
Height: hc = 25.24440543281

Median: ma = 25.95986283432
Median: mb = 21.87332230241
Median: mc = 25.76999719313

Inradius: r = 8.03545438078
Circumradius: R = 16.54997268103

Vertex coordinates: A[27.03215698938; 0] B[0; 0] C[8.6966420015; 25.24440543281]
Centroid: CG[11.90993299696; 8.4154684776]
Coordinates of the circumscribed circle: U[13.51657849469; 9.46438545046]
Coordinates of the inscribed circle: I[11.26657849469; 8.03545438078]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.9911488279° = 125°59'29″ = 0.94326263536 rad
∠ B' = β' = 109.0098511721° = 109°31″ = 1.23990352114 rad
∠ C' = γ' = 125° = 0.96599310886 rad

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How did we calculate this triangle?

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

a = 26.7 ; ; b = 31.2 ; ; gamma = 55° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 26.7**2+31.2**2 - 2 * 26.7 * 31.2 * cos 55° } ; ; c = 27.03 ; ;
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 = 26.7 ; ; b = 31.2 ; ; c = 27.03 ; ;

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

p = a+b+c = 26.7+31.2+27.03 = 84.93 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 84.93 }{ 2 } = 42.47 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42.47 * (42.47-26.7)(42.47-31.2)(42.47-27.03) } ; ; T = sqrt{ 116412.81 } = 341.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 341.19 }{ 26.7 } = 25.56 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 341.19 }{ 31.2 } = 21.87 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 341.19 }{ 27.03 } = 25.24 ; ;

6. 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{ 31.2**2+27.03**2-26.7**2 }{ 2 * 31.2 * 27.03 } ) = 54° 31" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 26.7**2+27.03**2-31.2**2 }{ 2 * 26.7 * 27.03 } ) = 70° 59'29" ; ; gamma = 180° - alpha - beta = 180° - 54° 31" - 70° 59'29" = 55° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 341.19 }{ 42.47 } = 8.03 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 26.7 }{ 2 * sin 54° 31" } = 16.5 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 31.2**2+2 * 27.03**2 - 26.7**2 } }{ 2 } = 25.959 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.03**2+2 * 26.7**2 - 31.2**2 } }{ 2 } = 21.873 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 31.2**2+2 * 26.7**2 - 27.03**2 } }{ 2 } = 25.7 ; ;
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