Triangle calculator SAS

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


Acute isosceles triangle.

Sides: a = 5.5   b = 5.5   c = 2.84770094961

Area: T = 7.56325
Perimeter: p = 13.84770094961
Semiperimeter: s = 6.92435047481

Angle ∠ A = α = 75° = 1.3098996939 rad
Angle ∠ B = β = 75° = 1.3098996939 rad
Angle ∠ C = γ = 30° = 0.52435987756 rad

Height: ha = 2.75
Height: hb = 2.75
Height: hc = 5.31325920446

Median: ma = 3.40881126061
Median: mb = 3.40881126061
Median: mc = 5.31325920446

Inradius: r = 1.09222936107
Circumradius: R = 2.84770094961

Vertex coordinates: A[2.84770094961; 0] B[0; 0] C[1.42435047481; 5.31325920446]
Centroid: CG[1.42435047481; 1.77108640149]
Coordinates of the circumscribed circle: U[1.42435047481; 2.46655825485]
Coordinates of the inscribed circle: I[1.42435047481; 1.09222936107]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105° = 1.3098996939 rad
∠ B' = β' = 105° = 1.3098996939 rad
∠ C' = γ' = 150° = 0.52435987756 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 = 5.5 ; ; b = 5.5 ; ; gamma = 30° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 5.5**2+5.5**2 - 2 * 5.5 * 5.5 * cos(30° ) } ; ; c = 2.85 ; ;


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 = 5.5 ; ; b = 5.5 ; ; c = 2.85 ; ;

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

p = a+b+c = 5.5+5.5+2.85 = 13.85 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 13.85 }{ 2 } = 6.92 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.92 * (6.92-5.5)(6.92-5.5)(6.92-2.85) } ; ; T = sqrt{ 57.19 } = 7.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.56 }{ 5.5 } = 2.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.56 }{ 5.5 } = 2.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.56 }{ 2.85 } = 5.31 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 5.5**2-5.5**2-2.85**2 }{ 2 * 5.5 * 2.85 } ) = 75° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.5**2-5.5**2-2.85**2 }{ 2 * 5.5 * 2.85 } ) = 75° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.85**2-5.5**2-5.5**2 }{ 2 * 5.5 * 5.5 } ) = 30° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.56 }{ 6.92 } = 1.09 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.5 }{ 2 * sin 75° } = 2.85 ; ;




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