ISEE Upper Level Quantitative : Solid Geometry
Study concepts, example questions & explanations for ISEE Upper Level Quantitative
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Example Questions
Example Question #1 :How To Find The Surface Area Of A Prism
(A) The surface area of a rectangular prism with length
(B) The surface area of a rectangular prism with length
It is impossible to determine which is greater from the information given
(A) is greater
(B) is greater
(A) and (B) are equal
(A) is greater
The surface area of a rectangular prism can be determined using the formula:
Using substitutions, the surface areas of the prisms can be found as follows:
The prism in (A):
Regardless of the value of
Example Question #301 :Geometry
一个立方体sidelength有脚;一个矩形tangular prism of equal volume has length 27 inches and height 9 inches. Give the width of the prism in inches.
One and one half feet is equal to eighteen inches, so the volume of the cube, in cubic inches, is the cube of this, or
The volume of a rectangular prism is
Since its volume is the same as that of the cube, and its length and height are 27 and 9 inches, respectively, we can rewrite this as
The width is 24 inches.
Example Question #2 :How To Find The Length Of An Edge
一个立方体sidelength有脚;一个矩形tangular prism of equal surface area has length 27 inches and height 9 inches. Give the width of the prism in inches.
One and one half feet is equal to eighteen inches, so the surface area of the cube, in square inches, is six times the square of this, or
The surface area of a rectangular prism is determined by the formula
So, with substitutiton, we can find the width:
Example Question #1 :Solid Geometry
A rectangular prism has volume one cubic foot; its length and width are, respectively, 9 inches and
The volume of a rectangular prism is the product of its length, its width, and its height. The prism's volume of one cubic foot is equal to
Therefore,
We can solve for
Example Question #1 :Find The Volume Of A Right Rectangular Prism With Fractional Edge Lengths: Ccss.Math.Content.6.G.A.2
一大箱形状的矩形棱镜has dimensions 5 feet by 4 feet by 12 feet. Give its volume incubic yards.
Divide each dimension by 3 to convert feet to yards, then multiply the three dimensions together:
Example Question #1 :Find The Volume Of A Right Rectangular Prism With Fractional Edge Lengths: Ccss.Math.Content.6.G.A.2
Which is the greater quantity?
(A) The volume of a rectangular solid ten inches by twenty inches by fifteen inches
(B) The volume of a cube with sidelength sixteen inches
It is impossible to determine which is greater from the information given
(A) is greater
(A) and (B) are equal
(B) is greater
(B) is greater
The volume of a rectangular solid ten inches by twenty inches by fifteen inches is
The volume of a cube with sidelength 13 inches is
This makes (B) greater
Example Question #1 :Pyramids
Pyramid 1 has a square base with sidelength
Pyramid 2 has a square base with sidelength
Which is the greater quantity?
(a) The volume of Pyramid 1
(b) The volume of Pyramid 2
It is impossible to tell from the information given.
(b) is greater.
(a) is greater.
(a) and (b) are equal.
(b) is greater.
Use the formula
(a)
(b)
Regardless of
Example Question #2 :Pyramids
Which is the greater quantity?
(a) The volume of a pyramid with height 4, the base of which has sidelength 1
(b) The volume of a pyramid with height 1, the base of which has sidelength 2
(a) is greater.
(b) is greater.
(a) and (b) are equal.
It is impossible to tell from the information given.
(a) and (b) are equal.
The volume of a pyramid with height
(a) Substitute
(b) Substitute
The two pyramids have equal volume.
Example Question #1 :Solid Geometry
Which is the greater quantity?
(a) The volume of a pyramid whose base is a square with sidelength 8 inches
(b) The volume of a pyramid whose base is an equilateral triangle with sidelength one foot
(a) and (b) are equal.
(b) is greater.
(a) is greater.
It is impossible to tell from the information given.
It is impossible to tell from the information given.
The volume of a pyramid is one-third of the product of the height and the area of the base. The areas of the bases can be calculated, but no information is given about the heights of the pyramids. There is not enough information to determine which one has the greater volume.
Example Question #1 :Pyramids
A pyramid with a square base has height equal to the perimeter of its base. Its volume is
The volume of a pyramid is given by the formula
where
Let
and the area of the base is
So the volume formula becomes
Solve for
All ISEE Upper Level Quantitative Resources
