Solids with known cross sections worksheet

Nothing spins here. The solid sits on a flat region and every slice through it is the same shape, so working out the area of one slice and integrating it along the base gives the volume.

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Sigma Prep · sigmaprep.io/worksheets Name Known Cross Sections Date Period Find the volume of each solid. Give exact values. Find volumes with known cross sections The base of a solid is the region under $y = \sqrt{x}$ from $x = 2$ to $x = 6$ . Its cross sections perpendicular to the $x$ -axis are equilateral triangles. Find the volume. Find the volume of the solid whose base is the region bounded by $y = \frac{x}{3}$ , the $y$ -axis and $y = 4$ and whose cross sections perpendicular to the $y$ -axis are semicircles on a diameter. A solid has as its base the triangle bounded by $y = 3 - x$ and the axes. Each slice perpendicular to the $x$ -axis is a square. Find the volume of the solid. A solid has as its base the region under $y = 2 x$ from $x = 1$ to $x = 2$ . Each slice perpendicular to the $x$ -axis is an isosceles right triangle with a leg in the base. Find the volume of the solid. The base of a solid is the region under $y = x$ from $x = 0$ to $x = 3$ . Its cross sections perpendicular to the $x$ -axis are semicircles on a diameter. Find the volume. Find the volume of the solid whose base is the triangle bounded by $y = 5 - x$ and the axes and whose cross sections perpendicular to the $x$ -axis are squares. A solid has as its base the region bounded by $y = \frac{x}{3}$ , the $y$ -axis and $y = 1$ . Each slice perpendicular to the $y$ -axis is an isosceles right triangle with a leg in the base. Find the volume of the solid. Find volumes, slicing perpendicular to the $y$ -axis A solid's base is the region between $y = x^{2}$ and $y = 6$ . Cross sections perpendicular to the $y$ -axis are semicircles. Find the volume. A solid's base is the region between $y = x^{2}$ and $y = 2$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume. A solid's base is the region between $y = x^{2}$ and $y = 3$ . Cross sections perpendicular to the $y$ -axis are semicircles. Find the volume. A solid's base is the region between $y = x^{2}$ and $y = 4$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume. A solid's base is the region between $y = x^{2}$ and $y = 5$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume. A solid's base is the region between $y = x^{2}$ and $y = 1$ . Cross sections perpendicular to the $y$ -axis are semicircles. Find the volume. A solid's base is the region between $y = x^{2}$ and $y = 9$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume. Find volumes with varied bases and cross sections A solid has base the disk $x^{2} + y^{2} \leq 49$ . Cross sections perpendicular to the $x$ -axis are squares. Find its volume. A solid has base the region between $y = 2 x$ and $y = x^{2}$ . Cross sections perpendicular to the $x$ -axis are squares. Find its volume. A solid has base the disk $x^{2} + y^{2} \leq 4$ . Cross sections perpendicular to the $x$ -axis are semicircles. Find its volume. A solid has base the region under $y = \frac{1}{x}$ from $x = 1$ to $x = 6$ . Cross sections perpendicular to the $x$ -axis are squares. Find its volume. A solid has base the triangle bounded by the axes and $y = 2 - x$ . Cross sections perpendicular to the $x$ -axis are semicircles. Find its volume. A solid has base the region under $y = \sqrt{x}$ from $x = 0$ to $x = 4$ . Cross sections perpendicular to the $x$ -axis are rectangles whose height is $2$ times the base. Find its volume. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Known Cross Sections

Date Period

Find the volume of each solid. Give exact values.

Find volumes with known cross sections
The base of a solid is the region under $y = \sqrt{x}$ from $x = 2$ to $x = 6$ . Its cross sections perpendicular to the $x$ -axis are equilateral triangles. Find the volume.
Find the volume of the solid whose base is the region bounded by $y = \frac{x}{3}$ , the $y$ -axis and $y = 4$ and whose cross sections perpendicular to the $y$ -axis are semicircles on a diameter.
A solid has as its base the triangle bounded by $y = 3 - x$ and the axes. Each slice perpendicular to the $x$ -axis is a square. Find the volume of the solid.
A solid has as its base the region under $y = 2 x$ from $x = 1$ to $x = 2$ . Each slice perpendicular to the $x$ -axis is an isosceles right triangle with a leg in the base. Find the volume of the solid.
The base of a solid is the region under $y = x$ from $x = 0$ to $x = 3$ . Its cross sections perpendicular to the $x$ -axis are semicircles on a diameter. Find the volume.
Find the volume of the solid whose base is the triangle bounded by $y = 5 - x$ and the axes and whose cross sections perpendicular to the $x$ -axis are squares.
A solid has as its base the region bounded by $y = \frac{x}{3}$ , the $y$ -axis and $y = 1$ . Each slice perpendicular to the $y$ -axis is an isosceles right triangle with a leg in the base. Find the volume of the solid.

Find volumes, slicing perpendicular to the $y$ -axis
A solid's base is the region between $y = x^{2}$ and $y = 6$ . Cross sections perpendicular to the $y$ -axis are semicircles. Find the volume.
A solid's base is the region between $y = x^{2}$ and $y = 2$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume.
A solid's base is the region between $y = x^{2}$ and $y = 3$ . Cross sections perpendicular to the $y$ -axis are semicircles. Find the volume.
A solid's base is the region between $y = x^{2}$ and $y = 4$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume.
A solid's base is the region between $y = x^{2}$ and $y = 5$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume.
A solid's base is the region between $y = x^{2}$ and $y = 1$ . Cross sections perpendicular to the $y$ -axis are semicircles. Find the volume.
A solid's base is the region between $y = x^{2}$ and $y = 9$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume.

Find volumes with varied bases and cross sections
A solid has base the disk $x^{2} + y^{2} \leq 49$ . Cross sections perpendicular to the $x$ -axis are squares. Find its volume.
A solid has base the region between $y = 2 x$ and $y = x^{2}$ . Cross sections perpendicular to the $x$ -axis are squares. Find its volume.
A solid has base the disk $x^{2} + y^{2} \leq 4$ . Cross sections perpendicular to the $x$ -axis are semicircles. Find its volume.
A solid has base the region under $y = \frac{1}{x}$ from $x = 1$ to $x = 6$ . Cross sections perpendicular to the $x$ -axis are squares. Find its volume.
A solid has base the triangle bounded by the axes and $y = 2 - x$ . Cross sections perpendicular to the $x$ -axis are semicircles. Find its volume.
A solid has base the region under $y = \sqrt{x}$ from $x = 0$ to $x = 4$ . Cross sections perpendicular to the $x$ -axis are rectangles whose height is $2$ times the base. Find its volume.



Sigma Prep · sigmaprep.io/worksheets Name Known Cross SectionsAnswer keyVersion 1 Date Period Find the volume of each solid. Give exact values. Find volumes with known cross sections The base of a solid is the region under $y = \sqrt{x}$ from $x = 2$ to $x = 6$ . Its cross sections perpendicular to the $x$ -axis are equilateral triangles. Find the volume. $4 \sqrt{3}$ Find the volume of the solid whose base is the region bounded by $y = \frac{x}{3}$ , the $y$ -axis and $y = 4$ and whose cross sections perpendicular to the $y$ -axis are semicircles on a diameter. $24 π$ A solid has as its base the triangle bounded by $y = 3 - x$ and the axes. Each slice perpendicular to the $x$ -axis is a square. Find the volume of the solid. $9$ A solid has as its base the region under $y = 2 x$ from $x = 1$ to $x = 2$ . Each slice perpendicular to the $x$ -axis is an isosceles right triangle with a leg in the base. Find the volume of the solid. $\frac{14}{3}$ The base of a solid is the region under $y = x$ from $x = 0$ to $x = 3$ . Its cross sections perpendicular to the $x$ -axis are semicircles on a diameter. Find the volume. $\frac{9 π}{8}$ Find the volume of the solid whose base is the triangle bounded by $y = 5 - x$ and the axes and whose cross sections perpendicular to the $x$ -axis are squares. $\frac{125}{3}$ A solid has as its base the region bounded by $y = \frac{x}{3}$ , the $y$ -axis and $y = 1$ . Each slice perpendicular to the $y$ -axis is an isosceles right triangle with a leg in the base. Find the volume of the solid. $\frac{3}{2}$ Find volumes, slicing perpendicular to the $y$ -axis A solid's base is the region between $y = x^{2}$ and $y = 6$ . Cross sections perpendicular to the $y$ -axis are semicircles. Find the volume. $9 π$ A solid's base is the region between $y = x^{2}$ and $y = 2$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume. $8$ A solid's base is the region between $y = x^{2}$ and $y = 3$ . Cross sections perpendicular to the $y$ -axis are semicircles. Find the volume. $\frac{9 π}{4}$ A solid's base is the region between $y = x^{2}$ and $y = 4$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume. $32$ A solid's base is the region between $y = x^{2}$ and $y = 5$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume. $50$ A solid's base is the region between $y = x^{2}$ and $y = 1$ . Cross sections perpendicular to the $y$ -axis are semicircles. Find the volume. $\frac{π}{4}$ A solid's base is the region between $y = x^{2}$ and $y = 9$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume. $162$ Find volumes with varied bases and cross sections A solid has base the disk $x^{2} + y^{2} \leq 49$ . Cross sections perpendicular to the $x$ -axis are squares. Find its volume. $\frac{5488}{3}$ A solid has base the region between $y = 2 x$ and $y = x^{2}$ . Cross sections perpendicular to the $x$ -axis are squares. Find its volume. $\frac{16}{15}$ A solid has base the disk $x^{2} + y^{2} \leq 4$ . Cross sections perpendicular to the $x$ -axis are semicircles. Find its volume. $\frac{16 π}{3}$ A solid has base the region under $y = \frac{1}{x}$ from $x = 1$ to $x = 6$ . Cross sections perpendicular to the $x$ -axis are squares. Find its volume. $\frac{5}{6}$ A solid has base the triangle bounded by the axes and $y = 2 - x$ . Cross sections perpendicular to the $x$ -axis are semicircles. Find its volume. $\frac{π}{3}$ A solid has base the region under $y = \sqrt{x}$ from $x = 0$ to $x = 4$ . Cross sections perpendicular to the $x$ -axis are rectangles whose height is $2$ times the base. Find its volume. $16$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Known Cross SectionsAnswer keyVersion 1

Date Period

Find the volume of each solid. Give exact values.

Find volumes with known cross sections
The base of a solid is the region under $y = \sqrt{x}$ from $x = 2$ to $x = 6$ . Its cross sections perpendicular to the $x$ -axis are equilateral triangles. Find the volume. $4 \sqrt{3}$
Find the volume of the solid whose base is the region bounded by $y = \frac{x}{3}$ , the $y$ -axis and $y = 4$ and whose cross sections perpendicular to the $y$ -axis are semicircles on a diameter. $24 π$
A solid has as its base the triangle bounded by $y = 3 - x$ and the axes. Each slice perpendicular to the $x$ -axis is a square. Find the volume of the solid. $9$
A solid has as its base the region under $y = 2 x$ from $x = 1$ to $x = 2$ . Each slice perpendicular to the $x$ -axis is an isosceles right triangle with a leg in the base. Find the volume of the solid. $\frac{14}{3}$
The base of a solid is the region under $y = x$ from $x = 0$ to $x = 3$ . Its cross sections perpendicular to the $x$ -axis are semicircles on a diameter. Find the volume. $\frac{9 π}{8}$
Find the volume of the solid whose base is the triangle bounded by $y = 5 - x$ and the axes and whose cross sections perpendicular to the $x$ -axis are squares. $\frac{125}{3}$
A solid has as its base the region bounded by $y = \frac{x}{3}$ , the $y$ -axis and $y = 1$ . Each slice perpendicular to the $y$ -axis is an isosceles right triangle with a leg in the base. Find the volume of the solid. $\frac{3}{2}$

Find volumes, slicing perpendicular to the $y$ -axis
A solid's base is the region between $y = x^{2}$ and $y = 6$ . Cross sections perpendicular to the $y$ -axis are semicircles. Find the volume. $9 π$
A solid's base is the region between $y = x^{2}$ and $y = 2$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume. $8$
A solid's base is the region between $y = x^{2}$ and $y = 3$ . Cross sections perpendicular to the $y$ -axis are semicircles. Find the volume. $\frac{9 π}{4}$
A solid's base is the region between $y = x^{2}$ and $y = 4$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume. $32$
A solid's base is the region between $y = x^{2}$ and $y = 5$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume. $50$
A solid's base is the region between $y = x^{2}$ and $y = 1$ . Cross sections perpendicular to the $y$ -axis are semicircles. Find the volume. $\frac{π}{4}$
A solid's base is the region between $y = x^{2}$ and $y = 9$ . Cross sections perpendicular to the $y$ -axis are squares. Find the volume. $162$

Find volumes with varied bases and cross sections
A solid has base the disk $x^{2} + y^{2} \leq 49$ . Cross sections perpendicular to the $x$ -axis are squares. Find its volume. $\frac{5488}{3}$
A solid has base the region between $y = 2 x$ and $y = x^{2}$ . Cross sections perpendicular to the $x$ -axis are squares. Find its volume. $\frac{16}{15}$
A solid has base the disk $x^{2} + y^{2} \leq 4$ . Cross sections perpendicular to the $x$ -axis are semicircles. Find its volume. $\frac{16 π}{3}$
A solid has base the region under $y = \frac{1}{x}$ from $x = 1$ to $x = 6$ . Cross sections perpendicular to the $x$ -axis are squares. Find its volume. $\frac{5}{6}$
A solid has base the triangle bounded by the axes and $y = 2 - x$ . Cross sections perpendicular to the $x$ -axis are semicircles. Find its volume. $\frac{π}{3}$
A solid has base the region under $y = \sqrt{x}$ from $x = 0$ to $x = 4$ . Cross sections perpendicular to the $x$ -axis are rectangles whose height is $2$ times the base. Find its volume. $16$

How to do these

The base is the region under $y = x$ from $x = 0$ to $x = 3$ , with square cross sections.

Each slice is a square whose side is the height of the region, which is x.
So its area is x squared.
Integrating that from 0 to 3 gives 9.

The answer is $9$ .

Where students go wrong

Using pi r squared for the semicircle with r equal to the height. The height is the diameter, not the radius, so the area of the slice is pi over eight times the height squared.

Also called cross sections, square cross sections, semicircular cross sections, equilateral triangle cross sections or volume by slicing.

What you can put on this worksheet

Solids with known cross sections: A solid has as its base the region under $y = 3 \sqrt{x}$ from $x = 0$ to $x = 2$ . Each slice perpendicular to the $x$ -axis is an isosceles right triangle with a leg in the base. Find the volume of the solid. → $9$
Perpendicular to the y-axis: A solid's base is the region between $y = x^{2}$ and $y = 5$ . Cross sections perpendicular to the $y$ -axis are semicircles. Find the volume. → $\frac{25 π}{4}$
Circular and parabolic bases, other shapes, across y: A solid has base the region between $y = 9 - x^{2}$ and the $x$ -axis. Cross sections perpendicular to the $x$ -axis are squares. Find its volume. → $\frac{1296}{5}$

Questions about these worksheets

Yes. Every solids with known cross sections sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the solids with known cross sections questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: solids with known cross sections, perpendicular to the y-axis and circular and parabolic bases, other shapes, across y. Tick as many as you want and set how many of each, or let it spread them evenly.

Using pi r squared for the semicircle with r equal to the height. The height is the diameter, not the radius, so the area of the slice is pi over eight times the height squared.

Solids with known cross sections worksheet

Select your difficulty

Find volumes with known cross sections

Find volumes, slicing perpendicular to the yyy-axis

Find volumes with varied bases and cross sections

Find volumes with known cross sections

Find volumes, slicing perpendicular to the yyy-axis

Find volumes with varied bases and cross sections

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these solids with known cross sections worksheets free?+

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What do students get wrong with solids with known cross sections?+

Find volumes, slicing perpendicular to the $y$ -axis

Find volumes, slicing perpendicular to the $y$ -axis