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Advanced Geometry : Solid Geometry

Study concepts, example questions & explanations for Advanced Geometry

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Example Questions

Example Question #1 : Advanced Geometry

What is the surface area of the following cone?

Possible Answers:

$\pi$ m^2

Correct answer:

$\pi$ m^2

Explanation:

The formula for the surface area of a cone is:

A = $\pi$ $\pi$ ,

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

Plugging in our values, we get:

A = $\pi$ $\pi$

A = $\pi$ m^2

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Example Question #1 : Cones

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?

Possible Answers:

$\small 9$

$\small 45$

$\small 3$

$\small 2\sqrt{5}$

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

Correct answer:

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

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 $\small 2\sqrt{14}$

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Example Question #2 : Advanced Geometry

A circle of radius five is cut into two pieces, $30\%$ and $70\%$ . 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?

Possible Answers:

$\frac{9\pi }{4}$

$2\pi$

$3\pi$

$25\pi$

Correct answer:

$\frac{9\pi }{4}$

Explanation:

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 $30\%$ 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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Example Question #3 : Advanced Geometry

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

Possible Answers:

$24\pi$

$20\pi$

$12\pi$

Correct answer:

$24\pi$

Explanation:

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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Example Question #4 : Advanced Geometry

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?

Possible Answers:

$16\pi$

Correct answer:

Explanation:

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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Example Question #5 : Advanced Geometry

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 $\pi$ ?

Possible Answers:

$130\pi$

$\small 200\pi$

$30\pi$

$\small 209\pi$

$\small 100\pi+10\pi\sqrt{109}$

Correct answer:

$\small 100\pi+10\pi\sqrt{109}$

Explanation:

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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Example Question #6 : Advanced Geometry

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?

Possible Answers:

$\small 5$

$\small 6$

$\small \small 10$

$\small 10\pi$

$\small 3$

Correct answer:

$\small 5$

Explanation:

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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Example Question #1 : Advanced Geometry

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

Possible Answers:

$72\pi$

$96\pi$

$30\pi$

$90\pi$

$39\pi$

Correct answer:

$39\pi$

Explanation:

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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Example Question #2 : Advanced Geometry

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

Possible Answers:

$\frac{16\sqrt2}{3} \pi$

$24\pi$

$8\pi$

$8\sqrt2\pi +4\pi$

$16\pi$

Correct answer:

$16\pi$

Explanation:

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.

$A = \pi r^2$

$4\pi = \pi r^2$

4=r^2

2=r

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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Example Question #1 : Advanced Geometry

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

Possible Answers:

$39\pi$

$12\pi\sqrt{109}$

$3\pi\sqrt{109} +9\pi$

$3\pi \sqrt{109}$

$30\pi$

Correct answer:

$3\pi\sqrt{109} +9\pi$

Explanation:

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.

r^2 + h^2 = s^2

(3)^2 + (10)^2 = s^2

109= s^2

$\sqrt{109} = s$

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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Advanced Geometry : Solid Geometry

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #1 : Advanced Geometry

Example Question #1 : Cones

Example Question #2 : Advanced Geometry

Example Question #3 : Advanced Geometry

Example Question #4 : Advanced Geometry

Example Question #5 : Advanced Geometry

Example Question #6 : Advanced Geometry

Example Question #1 : Advanced Geometry

Example Question #2 : Advanced Geometry

Example Question #1 : Advanced Geometry

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