Triangle calculator SAS

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


Obtuse isosceles triangle.

Sides: a = 27.8   b = 27.8   c = 53.70554759417

Area: T = 193.21
Perimeter: p = 109.3055475942
Semiperimeter: s = 54.65327379708

Angle ∠ A = α = 15° = 0.26217993878 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 150° = 2.6187993878 rad

Height: ha = 13.9
Height: hb = 13.9
Height: hc = 7.19551694539

Median: ma = 40.43994494654
Median: mb = 40.43994494654
Median: mc = 7.19551694539

Inradius: r = 3.53552300209
Circumradius: R = 53.70554759417

Vertex coordinates: A[53.70554759417; 0] B[0; 0] C[26.85327379708; 7.19551694539]
Centroid: CG[26.85327379708; 2.3988389818]
Coordinates of the circumscribed circle: U[26.85327379708; -46.51103064878]
Coordinates of the inscribed circle: I[26.85327379708; 3.53552300209]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165° = 0.26217993878 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 30° = 2.6187993878 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 = 27.8 ; ; b = 27.8 ; ; gamma = 150° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 27.8**2+27.8**2 - 2 * 27.8 * 27.8 * cos(150° ) } ; ; c = 53.71 ; ;


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 = 27.8 ; ; b = 27.8 ; ; c = 53.71 ; ;

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

p = a+b+c = 27.8+27.8+53.71 = 109.31 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 109.31 }{ 2 } = 54.65 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 54.65 * (54.65-27.8)(54.65-27.8)(54.65-53.71) } ; ; T = sqrt{ 37330.1 } = 193.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 193.21 }{ 27.8 } = 13.9 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 193.21 }{ 27.8 } = 13.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 193.21 }{ 53.71 } = 7.2 ; ;

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{ 27.8**2+53.71**2-27.8**2 }{ 2 * 27.8 * 53.71 } ) = 15° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 27.8**2+53.71**2-27.8**2 }{ 2 * 27.8 * 53.71 } ) = 15° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 27.8**2+27.8**2-53.71**2 }{ 2 * 27.8 * 27.8 } ) = 150° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 193.21 }{ 54.65 } = 3.54 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 27.8 }{ 2 * sin 15° } = 53.71 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.8**2+2 * 53.71**2 - 27.8**2 } }{ 2 } = 40.439 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 53.71**2+2 * 27.8**2 - 27.8**2 } }{ 2 } = 40.439 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.8**2+2 * 27.8**2 - 53.71**2 } }{ 2 } = 7.195 ; ;
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