ISEE Upper Level Quantitative : Solid Geometry

Example Questions

Example Question #321 : Geometry

What is the length of one side of a cube that has a surface area of  ?

Explanation:

Recall that the formula for the surface area of a cube is:

, where  is the length of a side of the cube.  This equation is easy to memorize because it is merely a multiplication of a single side () by  because a cube has  equal sides.

Now, we know that  is ; therefore, we can write:

Solve for :

Take the square root of both sides:

This is the length of one of your sides.

Example Question #1 : Tetrahedrons

Which is the greater quantity?

(a) The surface area of a regular tetrahedron with edges of length 1

(b) 2

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

(b) is greater.

Explanation:

A regular tetrahedron has four faces, each of which is an equilateral triangle. Therefore, its surface area, given sidelength , is

.

Substitute  :

, so (b) is greater.

Example Question #1 : Spheres

The volume of a sphere is one cubic yard. Give its radius in inches.

Explanation:

The volume  of a sphere with radius  is

.

To find the radius in yards, we set  and solve for .

yards.

Since the problem requests the radius in inches, multiply by 36:

Example Question #1 : How To Find The Volume Of A Sphere

In terms of , give the volume, in cubic feet, of a spherical tank with diameter 36 inches.

Explanation:

36 inches =  feet, the diameter of the tank. Half of this, or  feet, is the radius. Set , substitute in the volume formula, and solve for :

Example Question #2 : How To Find The Volume Of A Sphere

Which is the greater quantity?

(a) The volume of a sphere with radius

(b) The volume of a cube with sidelength

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(b) is greater

Explanation:

A sphere with radius  has diameter  and can be inscribed inside a cube of sidelength . Therefore, the cube in (b) has the greater volume.

Example Question #31 : Solid Geometry

Which is the greater quantity?

(a) The volume of a cube with sidelength  inches.

(b) The volume of a sphere with radius  inches.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

Explanation:

You do not need to calculate the volumes of the figures. All you need to do is observe that a sphere with radius  inches has diameter  inches, and can therefore be inscribed inside the cube with sidelength  inches. This give the cube larger volume, making (a) the greater quantity.

Example Question #3 : How To Find The Volume Of A Sphere

Which is the greater quantity?

(a) The volume of a sphere with diameter one foot

(b)

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

(a) is greater.

Explanation:

The radius of the sphere is one half of its diameter of one foot, which is six inches, so substitute :

cubic inches,

which is greater than .

Example Question #1 : Spheres

is a positive number. Which is the greater quantity?

(A) The volume of a cube with edges of length

(B) The volume of a sphere with radius

(A) is greater

It is impossible to determine which is greater from the information given

(B) is greater

(A) and (B) are equal

(A) is greater

Explanation:

No calculation is really needed here, as a sphere with radius  - and, subsequently, diameter  - can be inscribed inside a cube of sidelength . This makes (A), the volume of the cube, the greater.

Example Question #5 : How To Find The Volume Of A Sphere

Which is the greater quantity?

(a) The radius of a sphere with surface area

(b) The radius of a sphere with volume

(a) and (b) are equal

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(b) is the greater quantity

(a) and (b) are equal

Explanation:

The formula for the surface area of a sphere, given its radius , is

The sphere in (a) has surface area , so

The formula for the volume of a sphere, given its radius , is

The sphere in (b) has volume , so

The radius of both spheres is 3.

Example Question #1 : How To Find The Surface Area Of A Sphere

In terms of , give the surface area, in square inches, of a spherical water tank with a diameter of 20 feet.