Cones - Advanced Geometry

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Question

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)?

Answer

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

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

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Question

A right cone has a radius of 4R and a height of 3R. What is the ratio of the total surface area of the cone to the surface area of just the base?

Answer

We need to find total surface area of the cone and the area of the base.

The area of the base of a cone is equal to the area of a circle. The formula for the area of a circle is given below:

$A=\pi r^2$ , where r is the length of the radius.

In the case of this cone, the radius is equal to 4R, so we must replace r with 4R.

$A=\pi(4R)^2=\pi(4)^2(R)^2=16\pi R^2$

To find the total area of the cone, we need the area of the base and the lateral surface area of the cone. The lateral surface area (LA) of a cone is given by the following formula:

$LA=\frac{1}{2}(2\pi r)(l)$ , where r is the radius and l is the slant height.

We know that r = 4R. What we need now is the slant height, which is the distance from the edge of the base of the cone to the tip.

In order to find the slant height, we need to construct a right triangle with the legs equal to the height and the radius of the cone. The slant height will be the hypotenuse of this triangle. We can use the Pythagorean Theorem to find an expression for l. According to the Pythagorean Theorem, the sum of the squares of the legs (which are 4R and 3R in this case) is equal to the square of the hypotenuse (which is the slant height). According to the Pythagorean Theorem, we can write the following equation:

Let's go back to the formula for the lateral surface area (LA).

$LA=\frac{1}{2}(2\pi r)(l)$

$LA=\frac{1}{2}(2\pi(4R))(5R)$

$LA=20\pi R^2$

To find the total surface area (TA), we must add the lateral area and the area of the base.

$TA=20\pi R^2+16\pi R^2=36\pi R^2$

The problem requires us to find the ratio of the total surface area to the area of the base. This means we must find the following ratio:

$\frac{36\pi R^2}{16\pi R^2}$

We can cancel $\pi R^2$ , which leaves us with 36/16.

Simplifying 36/16 gives 9/4.

The answer is 9/4.

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Question

What is the surface area of a cone with a radius of 6 in and a height of 8 in?

Answer

Find the slant height of the cone using the Pythagorean theorem: _r_2 + _h_2 = _s_2 resulting in 62 + 82 = _s_2 leading to _s_2 = 100 or s = 10 in

SA = πrs + πr_2 = π(6)(10) + π(6)2 = 60_π + 36_π_ = 96_π_ in2

60_π_ in2 is the area of the cone without the base.

36_π_ in2 is the area of the base only.

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Question

What is the surface area of a cone with a radius of 4 and a height of 3?

Answer

Here we simply need to remember the formula for the surface area of a cone and plug in our values for the radius and height.

$\Pi r^{2} + \Pi r\sqrt{r^{2} + h^{2}}= \Pi\ast 4^{2} + \Pi \ast 4\sqrt{4^{2} + 3^{2}} = 16\Pi + 4\Pi \sqrt{25} = 16\Pi + 20\Pi = 36\Pi$

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Question

What is the surface area of the following cone?

Answer

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:

$\pi$ $\pi$

$\pi$ $\pi$

$\pi$

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Question

If the surface area of a right angle cone $\small A$ is $\small 48\pi$ , and the distance from the tip of the cone to a point on the edge of the cone's base is $\small 8$ , what is the cone's radius?

Answer

Solving this problem is going to take knowledge of Algebra, Geometry, and the equation for the surface area of a cone: $\small \small A=\pi r^2 + \pi r s$ , where $\small r$ is the radius of the cone's base and $\small s$ is the distance from the tip of the cone to a point along the edge of the cone's base. First, let's substitute what we know in this equation:

$\small \small 48\pi=\pi r^2 + \pi r (8)=\pi r^2 + 8\pi r$

We can divide out $\small \small \pi$ from every term in the equation to obtain:

$\small \small \small 48=r^2+8r\Rightarrow0=r^2+8r-48$

We see this equation has taken the form of a quadratic expression, so to solve for $\small r$ we need to find the zeroes by factoring. We therefore need to find factors of $\small \small -48$ that when added equal $\small \small 8$ . In this case, $\small -4$ and $\small \small 12$ :

$\small \small r^2+8r-48=(r+12)(r-4)$

This gives us solutions of $\small \small r=-12$ and $\small r=4$ . Since $\small r$ represents the radius of the cone and the radius must be positive, we know that $\small r=4$ is our only possible answer, and therefore the radius of the cone is $\small 4$ .

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Question

For a right circular cone , the radius is and the height of the cone is . What is the surface area of the cone in terms of $\pi$ ?

Answer

To solve this problem, we will need to use the formula for finding the surface area of a cone, $\small A=\pi r^2+\pi rs$ , where $\small s$ is the length of the diagonal from the circle edge of the cone to the top. Since we are not given s, we must find it by using Pythagorean's Theorem:

$\small s=\sqrt{r^2+h^2}=\sqrt{10^2+3^2}=\sqrt{109}$ .

$\small 109$ is a prime number, so we cannot factor the radical any further. Therefore, our equation for our surface area of $\small C$ becomes:

$\small A=\pi(10^2)+\pi(10)\sqrt{109}=100\pi + 10\pi\sqrt{109}$ , which is our final answer.

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Question

If the surface area of cone $\small D$ is $\small 55\pi$ , and the distance $\small s$ between the cone's tip and a point on the cone's circular base is $\small 6$ , what is the radius $\small r$ of the cone?

Answer

To find out the radius, we must use our knowledge of the formula for the surface area of a cone: $\small A= \pi r^2 + \pi rs$ , where $\small r$ is the radius of the cone and $\small s$ is the distance from the tip of the cone to any point along the circumference of the cone's base. We can plug in what we already know into the above equation:

$\small 55\pi=\pi r^2+6\pi r$

We can divide out $\small \pi$ from each term to obtain:

$\small 55=r^2+6r\Rightarrow0=r^2+6r-55$

We now can recognize that the above is a quadratic expression, so to solve for $\small r$ we can find the zeroes of the equation by factoring. We need two numbers which will multiply to $\small -55$ but will add to $\small 6$ (in this case $\small 11$ and $\small -5$ ). Therefore, we can factor the above to the following:

$\small r^2+6r-55=(r+11)(r-5)$ .

Our two solutions are therefore $\small r=-11$ and $\small r=5$ . Since $\small r$ represents the radius of the base of the cone, it must be positive, and that leaves $\small 5$ as our one and only answer.

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Question

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?

Answer

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:

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

so

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

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 $\small 2\sqrt{14}$

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Question

Use the following formula to answer the question.

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

The slant height of a right circular cone is . The radius is , and the height is . Determine the surface area of the cone.

Answer

Notice that the height of the cone is not needed to answer this question and is simply extraneous information. We are told that the radius is , and the slant height is .

First plug these numbers into the equation provided.

$SA=\pi r^2+\pi rl=\pi (2)^2+\pi (2)(6)=4\pi cm^2+12\pi cm^2$

Then simplify by combining like terms.

$4\pi cm^2+12\pi cm^2=16\pi cm^2$

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Question

A circle of radius five is cut into two pieces, and . The larger section is thrown away. The smaller section is curled until the two straight edges meet, and a bottom is made for the cone.

What is the area of the bottom?

Answer

When the smaller portion of the circle is curled in, it will make the top of a cone. The circumfrence of the circle on the bottom is $2\pi r$ (where r is the radius of the circle on the bottom). The circumference of the bottom is also of the circumfrence of the original larger circle, which is $.3C=.3(2\pi R)=.3(10\pi)=3\pi$ (where R is the radius of the original, larger circle)

Therefore we use the circumference formula to solve for our new r:

$2\pi r=3\pi, r=\frac{3}{2}$

Substituting this value into the area formula, the area of the small circle becomes:

$A=\pi r^2=\frac{9\pi}{4}$

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Question

A cone has a bottom area of $9\pi$ and a height of , what is the surface area of the cone?

Answer

The area of the bottom of the cone yields the radius,

$A=\pi r^2, r=\sqrt{\frac{A}{\pi}}=\sqrt{\frac{9\pi}{\pi}}=3$

The height of the cone is , so the Pythagorean Theorem will give the slant height, $l=\sqrt{3^2+4^2}=5$

The area of the side of the cone is $A=\pi rl=\pi \cdot 3 \cdot 5=15\pi$ and adding that to the $9\pi$ given as the area of the circle, the surface area comes to $24\pi$

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Question

Find the surface area of a cone with a base diameter of and a slant height of .

Answer

The Surface Area of a cone is:

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

Given the base diameter is 6, the radius will be 3. The given slant height is 10.

Substitute the radius and slant height into the equation to find surface area.

$SA=\pi (3)(10) +\pi (3)^2 = 30\pi + 9\pi = 39\pi$

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Question

Find the surface area of a cone with a base area of $4\pi$ and a slant height of .

Answer

The surface area of a cone is:

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

Given the base area is $4\pi$ , the base of the cone resembles a circle. Using the base area, it is necessary to find the radius.

$A = \pi r^2$

$4\pi = \pi r^2$

Since radius of the base is 2, and slant height is 6, substitute these into the surface area equation.

$SA= \pi (2)(6) + \pi (2)^2=12\pi+4\pi=16\pi$

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Question

Find the surface area of a cone with a base diameter of and a height of .

Answer

The Surface Area of a cone is:

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

Given the base diameter is 6, the radius of the base is 3. The height is 10. We will substitute these values to find the slant height by using the Pythagorean Theorem.

$\sqrt{109} = s$

Substitute slant height and radius into the Surface Area equation.

$SA=\pi (3)(\sqrt{109}) + \pi (3)^2 = 3\pi \sqrt{109} + 9\pi$

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Question

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 .

Answer

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:

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

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

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

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

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Question

The height of a cone is ; the diameter of its base is twice the height. Give its surface area in terms of .

Answer

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 twice the height, which is ; the radius is half this, which is .

The slant height can be calculated using the Pythagorean Theorem:

$l = \sqrt{r^{2}+ h^{2}}$

$l = \sqrt{h^{2}+ h^{2}}$

$l = \sqrt{2h^{2}}$

$l =h \sqrt{2}$

Substitute $h \sqrt{2}$ for and for in the surface area formula:

$A = \pi h^{2} + \pi h \cdot h \sqrt{2}$

$A = \pi h^{2} + \pi h ^{2} \sqrt{2}$

$A = \left (1 + \sqrt{2} \right ) \pi h^{2}$

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Question

The radius of the base of a cone is ; its height is twice of the diameter of that base. Give its surface area in terms of .

Answer

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 base has radius and diameter . The height is twice the diamter, which is . Its slant height can be calculated using the Pythagorean Theorem:

$l = \sqrt{r^{2}+ h^{2}}$

$l = \sqrt{r^{2}+ (4r)^{2}}$

$l = \sqrt{r^{2}+ 16r^{2}}$

$l = \sqrt{17r^{2}}$

$l = r \sqrt{17 }$

Substitute $r \sqrt{17 }$ for in the surface area formula:

$A = \pi r^{2} + \pi r \cdot r \sqrt{17}$

$A = \pi r^{2} + \pi r^{2} \sqrt{17}$

$A = \left ( 1+ \sqrt{17} \right ) \pi r^{2}$

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Question

The circumference of the base of a cone is 100; the height of the cone is equal to the diameter of the base. Give the surface area of the cone (nearest whole number).

Answer

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 the circumference divided by $\pi$ , which is

$d = C \div (2\pi ) \approx 100 \div 3.14 \approx 31.83$

This is also the height .

The radius is half this, or

$r = d \div 2 \approx 31.83 \div 2 \approx 15.92$

The slant height can be found by way of the Pythagorean Theorem:

$l = \sqrt{r^{2}+ h ^{2}}$

$l \approx \sqrt{15.92^{2}+ 31.83 ^{2}}$

$l \approx \sqrt{253.30+ 1,013.21}$

$l \approx \sqrt{1266.51}$

$l \approx 35.60$

Substitute in the surface area formula:

$A = \pi r^{2} + \pi r l$

$A \approx 3.14 \cdot 15.92 ^{2} + 3.14 \cdot 15.92 \cdot 35.60$

$A \approx 795.8+ 1,780.1$

$A \approx 2,575$

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Question

The circumference of the base of a cone is 80; the slant height of the cone is equal to twice the diameter of the base. Give the surface area of the cone (nearest whole number).

Answer

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 slant height is twice the diameter, or, equivalently, four times the radius, so

and

$A = \pi r^{2} + \pi r \cdot 4r$

$A = \pi r^{2} +4 \pi r^{2}$

$A = 5 \pi r^{2}$

The radius of the base is the circumference divided by $2\pi$ , which is

$80 \div 2 \pi \approx 80 \div (2 \times 3.14 )\approx 12.73$

Substitute:

$A = 5 \pi r^{2}$

$A \approx 5 \cdot 3.14 \cdot (12.73)^{2}$

$A \approx 2,546$

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