Surface Area of a Cone

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The surface area of a cone is equal to the curved surface area plus the area of the base: \(\pi r^2 + \pi L r, \) where \(r\) denotes the radius of the base of the cone, and \(L\) denotes the slant height of the cone. The curved surface area is also called the lateral area.

In the cone below, \(h\) is the height, \(L\) is the slant height, \(c\) is the circumference of the base, and \(r\) is the radius of the base. How does it look when we unroll it?

If we unroll it, the shape is as follows: It is a sector of a circle with radius \(L\) and arc length \(c\). So the curved surface area of the cone is the area of the sector above. The area of a sector given the arc length \(c\) and radius \(L\) is given by \(A=\dfrac{1}{2}cL\). Now applying this to the cone, we have \(A=\frac{1}{2}cL,\) where \(L\) is the slant height and \(c\) is the circumference of the base. After some manipulations, \(A=\pi Lr,\) as given in the definition.

A cone was formed by rolling a thin sheet of metal in the form of a sector of a circle with diameter \(72\text{ cm}\) and central angle \(150^\circ\). Find the curved surface area of the cone formed.

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The curved surface area is the area of the sector of a circle, so we have

\[A=\dfrac{\theta}{360}(\text{area of the circle})=\dfrac{150}{360}(\pi)\big(36^2\big)=540 \pi.\ _\square\]

A pile of sand is in the shape of a right circular cone. It has a height of \(3\) feet and a base diameter of \(8\) feet. Find its surface area in square feet.

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After some manipulations, we can see that a 3-4-5 right triangle is formed. Therefore, the slant height is \(5\) feet. So,

\[A=\pi r^2+\pi L r=\pi \big(4^2\big) +\pi(5)(4)=36 \pi.\ _\square\]

What is the lateral area of a cone that has a height of \(8\) and a radius of \(5?\)

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The first step is to compute the slant height \(L.\) Applying Pythagorean theorem, we have

\[L=\sqrt{8^2+5^2}=\sqrt{64+25}=\sqrt{89}.\]

Applying the formula, we have

\[A= \pi Lr=\pi \big(\sqrt{89}\big)(5)=5 \pi \sqrt{89}.\ _\square\]