Geometry

Help Questions

ISEE Upper Level: Quantitative Reasoning › Geometry

Questions 1 - 10

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:

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

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

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

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

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:

Trapezoidr

Find the area of the above trapezoid if $a=12\ cm$ , $b=14\ cm$ , and $h=10\ cm$ .

Figure not drawn to scale.

$130\ cm^2$

$120\ cm^2$

$110\ cm^2$

$100\ cm^2$

$90\ cm^2$

Explanation

The area of a trapezoid is given by

$Area=(\frac{b_{1}+b_{2}}{2})h$ ,

where , are the lengths of each base and is the altitude (height) of the trapezoid.

$b_{1}=a=12\ cm$

$b_{2}=b=14\ cm$

$h=10\ cm$

$Area=(\frac{b_{1}+b_{2}}{2})h=(\frac{12+14}{2})\times 10=13\times 10=130\ cm^2$

Find the surface area of a sphere with a diameter of 20in.

$400\pi \text{ in}^2$

$200\pi \text{ in}^2$

$800\pi \text{ in}^2$

$600\pi \text{ in}^2$

$300\pi \text{ in}^2$

Explanation

To find the surface area of a sphere, we will use the following formula:

$SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 20in. We also know the diameter is two times the radius. Therefore, the radius is 10in.

Knowing this, we can substitute into the formula. We get

$SA = 4 \cdot \pi \cdot (10\text{in})^2$

$SA = 4 \cdot \pi \cdot 100\text{in}^2$

$SA = 400 \pi \text{ in}^2$

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:

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

Find the perimeter of a pentagon with a side of length 12in.

$60\text{in}$

$72\text{in}$

$96\text{in}$

$48\text{in}$

$58\text{in}$

Explanation

To find the perimeter of a pentagon, we will use the following formula:

$P = 5 \cdot a$

where a is the length of one side of the pentagon.

Now, we know the length of one side of the pentagon is 12in.

Knowing this, we can substitute into the formula. We get

$P = 5 \cdot 12\text{in}$

$P = 60\text{in}$

Find the perimeter of a pentagon with a side of length 12in.

$60\text{in}$

$72\text{in}$

$96\text{in}$

$48\text{in}$

$58\text{in}$

Explanation

To find the perimeter of a pentagon, we will use the following formula:

$P = 5 \cdot a$

where a is the length of one side of the pentagon.

Now, we know the length of one side of the pentagon is 12in.

Knowing this, we can substitute into the formula. We get

$P = 5 \cdot 12\text{in}$

$P = 60\text{in}$

Icecreamcone

Refer to the above figure. The shaded region is a semicircle with area $16 \pi$ . Give the perimeter of $\bigtriangleup ABC$ .

$24 \sqrt{2}$

$24 \sqrt{3}$

$12 \sqrt{2}$

$12 \sqrt{3}$

Explanation

Given the radius of a semicircle, its area can be calculated using the formula

$A = \frac{1}{2} \pi r ^{2}$ .

Substituting $A = 16 \pi$ :

$\frac{1}{2} \pi r ^{2} = 16 \pi$

$\frac{2}{\pi} \cdot \frac{1}{2} \pi r ^{2} = \frac{2}{\pi} \cdot 16 \pi$

$r ^{2} = 32$

$r = \sqrt{32} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}$

The diameter of this semicircle is twice this, which is $d = 2 \cdot 4\sqrt{2} = 8 \sqrt{2}$ ; this is also the length of $\overline{BC}$ .

$\bigtriangleup ABC$ has two angles of degree measure 60; its third angle must also have measure 60, making $\bigtriangleup ABC$ an equilateral triangle with sidelength $s = 8 \sqrt{2}$ . Its perimeter is three times this, or $3 \cdot 8 \sqrt{2} = 24 \sqrt{2}$

Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the length of one side of her room?

Explanation

Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the length of one side of her room?

Let's begin with our formula for the area of a square: