Triangle calculator SAS

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


Acute isosceles triangle.

Sides: a = 54   b = 54   c = 25.21220992964

Area: T = 661.918814862
Perimeter: p = 133.2122099296
Semiperimeter: s = 66.60660496482

Angle ∠ A = α = 76.5° = 76°30' = 1.33551768778 rad
Angle ∠ B = β = 76.5° = 76°30' = 1.33551768778 rad
Angle ∠ C = γ = 27° = 0.4711238898 rad

Height: ha = 24.51554869859
Height: hb = 24.51554869859
Height: hc = 52.50879757015

Median: ma = 32.35546747081
Median: mb = 32.35546747081
Median: mc = 52.50879757015

Inradius: r = 9.93878082339
Circumradius: R = 27.7677210229

Vertex coordinates: A[25.21220992964; 0] B[0; 0] C[12.60660496482; 52.50879757015]
Centroid: CG[12.60660496482; 17.50326585672]
Coordinates of the circumscribed circle: U[12.60660496482; 24.74107654725]
Coordinates of the inscribed circle: I[12.60660496482; 9.93878082339]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103.5° = 103°30' = 1.33551768778 rad
∠ B' = β' = 103.5° = 103°30' = 1.33551768778 rad
∠ C' = γ' = 153° = 0.4711238898 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 = 54 ; ; b = 54 ; ; gamma = 27° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 54**2+54**2 - 2 * 54 * 54 * cos(27° ) } ; ; c = 25.21 ; ;


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 = 54 ; ; b = 54 ; ; c = 25.21 ; ;

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

p = a+b+c = 54+54+25.21 = 133.21 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 133.21 }{ 2 } = 66.61 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 66.61 * (66.61-54)(66.61-54)(66.61-25.21) } ; ; T = sqrt{ 438135.64 } = 661.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 661.92 }{ 54 } = 24.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 661.92 }{ 54 } = 24.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 661.92 }{ 25.21 } = 52.51 ; ;

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{ 54**2-54**2-25.21**2 }{ 2 * 54 * 25.21 } ) = 76° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 54**2-54**2-25.21**2 }{ 2 * 54 * 25.21 } ) = 76° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25.21**2-54**2-54**2 }{ 2 * 54 * 54 } ) = 27° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 661.92 }{ 66.61 } = 9.94 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 54 }{ 2 * sin 76° 30' } = 27.77 ; ;




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