A section is drawn through the top of the cone and the chord of the base, which contracts an arc of 120

| June 11, 2021 | education

A section is drawn through the top of the cone and the chord of the base, which contracts an arc of 120 degrees, making an angle of 45 degrees with the plane of the base. Find the area of the section if the radius of the base is 4 cm.

From the center of the circle O we draw the radii OA and OB, and the height OH.

The central angle BOH = 120, since it rests on an arc with a degree measure of 120, then the angle BOH = 10/2 = 60, and the angle OBH = 180 – 90 – 60 = 30.

In a right-angled triangle OBН, the OH leg lies opposite an angle of 300 and is equal to half the hypotenuse OB.

OH = OB / 2 = R / 2 = 4/2 = 2 cm.

Segment BH = OB * Cos30 = 4 * √3 / 2 = 2 * √3 cm.

Then AB = 2 * BH = 2 * 2 * √3 = 4 * √3 cm.

In a right-angled triangle СОH, the angle ОHС, by condition, is equal to 45, then the hypotenuse СН = ОН / Cos45 = 2 / (√2 / 2) = 2 * √2 cm.

Determine the cross-sectional area.

Savs = AB * CH / 2 = 4 * √3 * 2 * √2 / 2 = 4 * √6 cm2.

Answer: The cross-sectional area is 4 * √6 cm2.



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