Triangle calculator SAS

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


Acute isosceles triangle.

Sides: a = 9.28   b = 9.28   c = 10.49991476682

Area: T = 40.17222328468
Perimeter: p = 29.05991476682
Semiperimeter: s = 14.53295738341

Angle ∠ A = α = 55.55° = 55°33' = 0.97695303995 rad
Angle ∠ B = β = 55.55° = 55°33' = 0.97695303995 rad
Angle ∠ C = γ = 68.9° = 68°54' = 1.20325318546 rad

Height: ha = 8.65878088032
Height: hb = 8.65878088032
Height: hc = 7.65224750611

Median: ma = 8.75547501894
Median: mb = 8.75547501894
Median: mc = 7.65224750611

Inradius: r = 2.7654859679
Circumradius: R = 5.62768331039

Vertex coordinates: A[10.49991476682; 0] B[0; 0] C[5.25495738341; 7.65224750611]
Centroid: CG[5.25495738341; 2.55108250204]
Coordinates of the circumscribed circle: U[5.25495738341; 2.02656419572]
Coordinates of the inscribed circle: I[5.25495738341; 2.7654859679]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.45° = 124°27' = 0.97695303995 rad
∠ B' = β' = 124.45° = 124°27' = 0.97695303995 rad
∠ C' = γ' = 111.1° = 111°6' = 1.20325318546 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 = 9.28 ; ; b = 9.28 ; ; gamma = 68° 54' ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 9.28**2+9.28**2 - 2 * 9.28 * 9.28 * cos(68° 54') } ; ; c = 10.5 ; ;


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 = 9.28 ; ; b = 9.28 ; ; c = 10.5 ; ;

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

p = a+b+c = 9.28+9.28+10.5 = 29.06 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29.06 }{ 2 } = 14.53 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.53 * (14.53-9.28)(14.53-9.28)(14.53-10.5) } ; ; T = sqrt{ 1613.81 } = 40.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 40.17 }{ 9.28 } = 8.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 40.17 }{ 9.28 } = 8.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 40.17 }{ 10.5 } = 7.65 ; ;

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{ 9.28**2+10.5**2-9.28**2 }{ 2 * 9.28 * 10.5 } ) = 55° 33' ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9.28**2+10.5**2-9.28**2 }{ 2 * 9.28 * 10.5 } ) = 55° 33' ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 9.28**2+9.28**2-10.5**2 }{ 2 * 9.28 * 9.28 } ) = 68° 54' ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 40.17 }{ 14.53 } = 2.76 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.28 }{ 2 * sin 55° 33' } = 5.63 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.28**2+2 * 10.5**2 - 9.28**2 } }{ 2 } = 8.755 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.5**2+2 * 9.28**2 - 9.28**2 } }{ 2 } = 8.755 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.28**2+2 * 9.28**2 - 10.5**2 } }{ 2 } = 7.652 ; ;
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