A circle is inscribed in the trapezoid ABCD, bases BC = 12, AD = 16, lateral side CD = 15.

univerkov | April 29, 2021 | education

A circle is inscribed in the trapezoid ABCD, bases BC = 12, AD = 16, lateral side CD = 15. Find the area of the circle.

Since a circle is inscribed in the trapezoid, the sum of the lengths of its bases is equal to the sum of the lengths of its lateral sides.

BC + AD = AB + CD.

AB = BC + AD – CD = 12 + 16 – 15 = 13 cm.

To determine the area of a trapezoid, we use Heron’s formula for a trapezoid.

S = ((a + b) / | a – b | * √ (p – a) * (p – b) * (p – a – c) * (p – a – d), where p is the semi-perimeter of the trapezoid, a, b, c, d – the lengths of the sides of the trapezoid.

P = (12 + 16 + 13 + 15) / 2 = 28 cm.

S = (16 + 12) / (16 – 12) * √12 * 16 * (-1) * (-3) = 7 * 24 = 168 cm2.

Also, the area of the trapezoid is:

Savsd = (BC + AD) * KH / 2.

KN = 2 * Savsd / (BC + AD) = 2 * 168/28 = 12 cm.

Then OH = OK = R = 12/2 = 6 cm.

Determine the area of the circle.

Scr = n * R ^ 2 = 36 * n cm2.

Answer: The area of a circle is 36 * π cm2.

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