A circle of radius r is inscribed in the trapezoid. Find the area of the trapezoid if the angles for the larger

| August 8, 2021 | education

A circle of radius r is inscribed in the trapezoid. Find the area of the trapezoid if the angles for the larger base are equal to α and β.

Consider a trapezoid ABCD.

If a circle inscribed in a trapezoid has a center O and its points of contact with the sides of the trapezoid K, L, M, N, then we have:

OK = OL = OM = ON = r, where r is the radius of the circle. We also have that KN = 2 * r.

Let’s draw perpendiculars from vertices B and C to the AD side: BE and CF.

Since the circle is inscribed in a trapezoid, you can see that:

AL = AK, BL = BN, CN = CM, DM = DK.

Then:

AD + BC = AK + DK + BN + CN = AL + DM + BL + CM = AB + CD.

Triangles ABE and DCF are rectangular. Therefore:

AB * sin (α) = BE = CA = CD * sin (β) = 2 * r.

Then the area of ​​the trapezoid is:

S = (AD + BC) * BE / 2 = (AB + CD) * 2 * r / 2 = (AB + CD) * r =

= 2 * r * (1 / sin (α) + 1 / sin (β)) * r = 2 * r ^ 2 * (sin (α) + sin (β)) / (sin (α) * sin (β ))



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