Solid Geometry

Help Questions

ISEE Upper Level Quantitative Reasoning › Solid Geometry

Questions 1 - 10

A cube has a side length of , what is the volume of the cube?

Explanation

A cube has a side length of , what is the volume of the cube?

To find the volume of a cube, use the following formula:

$V_{cube}=s^3$

Plug in our known side length and solve

$V_{cube}=(9mm)^3=729mm^3$

Making our answer:

Box

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the volume of the box.

$15x^{2}$

$4x^{2} + 30 x$

Explanation

A square has four sides of equal length, as seen in the diagram below.

Box

The volume of the solid is equal to the product of its length, width, and height, as follows:

$V = 15 \cdot 15 \cdot x =225 x$ .

A wooden ball has a surface area of $900\pi m^2$ .

What is its radius?

Cannot be determined from the information provided

Explanation

A wooden ball has a surface area of $900\pi m^2$ .

What is its radius?

Begin with the formula for surface area of a sphere:

$SA_{sphere}=4\pi r^2$

Now, plug in our surface area and solve with algebra:

$900 \pi m^2=4\pi r^2$

Get rid of the pi

Divide by 4

Square root both sides to get our answer:

The distance from one vertex of a perfectly cubic aquarium to its opposite vertex is 1.5 meters. Give the volume of the aquarium in liters.

1 cubic meter = 1,000 liters.

The correct answer is not given among the other responses.

$375 \textup{ L}$

$421\frac{7}{8} \textup{ L}$

$843\frac{3}{4} \textup{ L}$

$750\textup{ L}$

Explanation

Let be the length of one edge of the cube. By the three-dimensional extension of the Pythagorean Theorem,

$s^{2}+s^{2}+s^{2}=1.5^{2}$

$3s^{2}=2.25$

$s^{2} = 0.75 = \frac{3}{4}$

$s = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{\sqrt{4}}} = \frac{\sqrt{3}}{2}$ meters.

Cube this sidelength to get the volume:

$s ^{3 } =\left ( \frac{\sqrt{3}}{2} \right )^{3 }= \frac{\left (\sqrt{3}\right )^{3 }}{2^{3 } } = \frac{ 3\sqrt{3} }{8 }$ cubic meters.

To convert this to liters, multiply by 1,000:

$\frac{ 3\sqrt{3} }{8 } \cdot 1,000 = \frac{ 3\sqrt{3} }{8 } \cdot \frac{1,000}{1} = \frac{ 3\sqrt{3} }{1 } \cdot \frac{125}{1}= 375\sqrt{3}$ liters.

This is not among the given responses.

The length of a diagonal of one face of a cube is . Give the volume of the cube.

$\frac{N^{3} \sqrt{2} }{4}$

$3N^{3}$

$\frac{N^{3} \sqrt{2} }{2}$

$\frac{N^{3} \sqrt{3} }{9}$

$\frac{N^{3} \sqrt{3} }{3}$

Explanation

Since a diagonal of a square face of the cube is, each side of each square has length $\frac{N}{\sqrt{2}}=\frac{N\sqrt{2}}{2}$ .

Cube this to get the volume of the cube:

$\left (\frac{N\sqrt{2}}{2} \right )^{3}$

$= \frac{N^{3}\left ( \sqrt{2} \right )^{3} }{2^{3} }$

$= \frac{N^{3} \cdot 2 \sqrt{2} }{8}$

$= \frac{N^{3} \sqrt{2} }{4}$

There is a perfectly spherical weather balloon with a surface area of $900 \pi ft^2$ , what is its diameter?

Explanation

There is a perfectly spherical weather balloon with a surface area of $900 \pi ft^2$ , what is its diameter?

Begin with the formula for surface area of a sphere:

$SA_{sphere}=4 \pi r^2$

Now, set it equal to the given surface area and solve for r:

$900 \pi ft^2=4 \pi r^2$

First divide both sides by $4\pi$ .

Then square root both sides to get our radius:

Now, because the question is asking for our diameter and not our radius, we need to double our radius to get our answer:

There is a perfectly spherical weather balloon with a surface area of $900 \pi ft^2$ , what is its diameter?

Explanation

There is a perfectly spherical weather balloon with a surface area of $900 \pi ft^2$ , what is its diameter?

Begin with the formula for surface area of a sphere:

$SA_{sphere}=4 \pi r^2$

Now, set it equal to the given surface area and solve for r:

$900 \pi ft^2=4 \pi r^2$

First divide both sides by $4\pi$ .

Then square root both sides to get our radius:

Now, because the question is asking for our diameter and not our radius, we need to double our radius to get our answer:

The length of a diagonal of one face of a cube is . Give the volume of the cube.

$\frac{N^{3} \sqrt{2} }{4}$

$3N^{3}$

$\frac{N^{3} \sqrt{2} }{2}$

$\frac{N^{3} \sqrt{3} }{9}$

$\frac{N^{3} \sqrt{3} }{3}$

Explanation

Since a diagonal of a square face of the cube is, each side of each square has length $\frac{N}{\sqrt{2}}=\frac{N\sqrt{2}}{2}$ .

Cube this to get the volume of the cube:

$\left (\frac{N\sqrt{2}}{2} \right )^{3}$

$= \frac{N^{3}\left ( \sqrt{2} \right )^{3} }{2^{3} }$

$= \frac{N^{3} \cdot 2 \sqrt{2} }{8}$

$= \frac{N^{3} \sqrt{2} }{4}$

There is a perfectly spherical weather balloon with a surface area of $900 \pi ft^2$ , what is its diameter?

Explanation

There is a perfectly spherical weather balloon with a surface area of $900 \pi ft^2$ , what is its diameter?

Begin with the formula for surface area of a sphere:

$SA_{sphere}=4 \pi r^2$

Now, set it equal to the given surface area and solve for r:

$900 \pi ft^2=4 \pi r^2$

First divide both sides by $4\pi$ .

Then square root both sides to get our radius:

Now, because the question is asking for our diameter and not our radius, we need to double our radius to get our answer:

A wooden ball has a surface area of $900\pi m^2$ .

What is its radius?

Cannot be determined from the information provided

Explanation

A wooden ball has a surface area of $900\pi m^2$ .

What is its radius?

Begin with the formula for surface area of a sphere:

$SA_{sphere}=4\pi r^2$

Now, plug in our surface area and solve with algebra:

$900 \pi m^2=4\pi r^2$

Get rid of the pi

Divide by 4

Square root both sides to get our answer:

Page 1 of 37

Return to subject

AI TutorYour personal study buddy

Question of the DayDaily practice to build your skills

FlashcardsStudy and memorize key concepts

WorksheetsBuild custom practice quizzes

AI SolverStep-by-step problem solutions

Learn by ConceptMaster concepts step by step

Diagnostic TestsAssess your knowledge level

Practice TestsTest your skills with exam questions

GamesLearn through free games