The base of the isosceles trapezoid is 12cm and 26cm, and the lateral side is 11cm.

| August 7, 2021 | education

The base of the isosceles trapezoid is 12cm and 26cm, and the lateral side is 11cm. find the volume of the figure formed by rotating this trapezoid around its axis.

Let’s complete the drawing. Let ABCD be a given trapezoid, AB = 12 cm, CD = 26 cm, AD = BC = 11 cm.

When the trapezoid is rotated around its axis, a truncated cone is obtained.

The volume of the truncated cone is calculated by the formula:

V = 1/3 * n * h * (R1² + R1R2 + R2²).

The radii of the base of the cone are equal to half of the base of the trapezoid: R1 = 6 cm, R2 = 13 cm.

Let’s draw the heights BH and AE, the quadrangle ABNE is a rectangle, so AB = EH. Hence CH = (26 – 12): 2 = 7 cm.

The triangle BHC is rectangular (BH is perpendicular to CD), we find BH by the Pythagorean theorem:

BH = √ (BC² – CH²) = √ (121 – 49) = √72 = 6√2 cm.

BH is not only the height of the trapezoid, but also the height of the truncated cone.

Let’s calculate the volume of the resulting cone:

V = 1/3 * n * 6√2 * (36 + 6 * 13 + 169) = 2√2p * 283 = 566p√2.



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