How to find the surface area of a cone

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Math › How to find the surface area of a cone

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The slant height of a cone is ; the diameter of its base is one-fifth its slant height. Give the surface area of the cone in terms of .

$\frac{11\pi L^{2}}{100}$

$\frac{6\pi L^{2}}{25}$

$\frac{ \pi L^{2}}{10}$

$\frac{ \pi L^{2}}{5}$

$\frac{ \pi L^{2}}{20}$

Explanation

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r L$ .

The diameter of the base is $\frac{1}{5}L$ ; the radius is half this, so

$r = \frac{1}{2} \cdot \frac{1}{5}L= \frac{1}{10}L$

Substitute in the surface area formula:

$A = \pi \left ( \frac{1}{10}L \right ) ^{2} + \pi \left ( \frac{1}{10}L \right ) L$

$A = \frac{1}{100} \pi L^{2} + \frac{1}{10} \pi L^{2}$

$A = \frac{1}{100} \pi L^{2} + \frac{10}{100} \pi L^{2}$

$A = \frac{11}{100} \pi L^{2} = \frac{11\pi L^{2}}{100}$

The surface area of cone $\small E$ is $\small 70\pi$ . If the radius of the base of the cone is , what is the height of the cone?

$\small \small 2\sqrt{14}$

$\small 45$

$\small 9$

$\small 3$

$\small 2\sqrt{5}$

Explanation

To figure out $\small h$ , we must use the equation for the surface area of a cone, $\small A=\pi r^2+\pi rs$ , where $\small r$ is the radius of the base of the cone and $\small s$ is the length of the diagonal from the tip of the cone to any point on the base's circumference. We therefore first need to solve for $\small s$ by plugging what we know into the equation:

$\small 70\pi=(5^2)\pi +5s\pi=25\pi +5s\pi$

This equation can be reduced to:

$\small 45\pi=5s\pi$

$\small 9=s$

For a normal right angle cone, $\small s$ represents the line from the tip of the cone running along the outside of the cone to a point on the base's circumference. This line represents the hypotenuse of the right triangle formed by the radius and height of the cone. We can therefore solve for $\small h$ using the Pythagorean theorem:

$\small s^2 = r^2+h^2$

$\small h=\sqrt{s^2-r^2}$

Our $\small h$ is therefore:

$\small h=\sqrt{9^2-5^2}=\sqrt{81-25}=\sqrt{56}=2\sqrt{14}$

The height of cone $\small E$ is therefore $\small 2\sqrt{14}$

What is the surface area of the following cone?

$\pi$

Explanation

The formula for the surface area of a cone is:

$\pi$ $\pi$ ,

where represents the radius of the cone base and represents the slant height of the cone.

Plugging in our values, we get:

$\pi$ $\pi$

$\pi$

Find the surface area of the following cone.

Cone

$90 \pi m^2$

$95 \pi m^2$

$100 \pi m^2$

$85 \pi m^2$

$80 \pi m^2$

Explanation

The formula for the surface area of a cone is:

$SA = A_{base} + A_{lateral}$

$SA = \pi r^2 + \pi r l$

where is the radius of the cone and is the slant height of the cone.

Plugging in our values, we get:

$SA = \pi (5m)^2 + \pi (5m) (13m)$

$SA = 25 \pi m^2 + 65 \pi m^2 = 90 \pi m^2$

The lateral area is twice as big as the base area of a cone. If the height of the cone is 9, what is the entire surface area (base area plus lateral area)?

27π

54π

81π

9π

90π

Explanation

Lateral Area = LA = π(r)(l) where r = radius of the base and l = slant height

LA = 2B

π(r)(l) = 2π(r2)

rl = 2r2

l = 2r

Cone

From the diagram, we can see that r2 + h2 = l2. Since h = 9 and l = 2r, some substitution yields

r2 + 92 = (2r)2

r2 + 81 = 4r2

81 = 3r2

27 = r2

B = π(r2) = 27π

LA = 2B = 2(27π) = 54π

SA = B + LA = 81π

Find the surface area of the following cone.

Cone

$96 \pi m^2$

$86 \pi m^2$

$76 \pi m^2$

$106 \pi m^2$

$116 \pi m^2$

Explanation

The formula for the surface area of a cone is:

$SA = (base) + \frac{(circumference)(slant height)}{2}$

$SA = (\pi r^2) + \frac{(2 \pi r)(h_s)}{2}$

$SA = (\pi r^2) + ( \pi rh_s)$

Use the Pythagorean Theorem to find the length of the radius:

Plugging in our values, we get:

$SA = \pi (6m)^2 + \pi (6m)(10m)$

$SA = 96 \pi m^2$

What is the surface area of a cone with a height of 8 and a base with a radius of 5?

$62.72\pi$

$25\pi$

$65\pi$

$48.45\pi$

Explanation

To find the surface area of a cone we must plug in the appropriate numbers into the equation

$Surface\ Area=\pi r^{2}+\pi rl$

where is the radius of the base, and is the lateral, or slant height of the cone.

First we must find the area of the circle.

To find the area of the circle we plug in our radius into the equation of a circle which is

This yields $A=25\pi$ .

We then need to know the surface area of the cone shape.

To find this we must use our height and our radius to make a right triangle in order to find the lateral height using Pythagorean’s Theorem.

Pythagorean’s Theorem states

Take the radius and height and plug them into the equation as a and b to yield $5^{2}+8^{2}=c^{2}$

First square the numbers $25+64=c^{2}$

After squaring the numbers add them together $89=c^{2}$

Once you have the sum, square root both sides $\sqrt{89}=\sqrt{c^{2}}$

After calculating we find our length is

Then plug the length into the second portion of our surface area equation above to get $(4)(\pi)(9.43)=37.72\pi$

Then add the area of the circle with the conical area to find the surface area of the entire figure $25\pi+37.72\pi=62.72\pi$

The answer is $62.72\pi$ .

The lateral area is twice as big as the base area of a cone. If the height of the cone is 9, what is the entire surface area (base area plus lateral area)?

27π

54π

81π

9π

90π