Extending to Three Dimensions
Unit 3 of Geometry. 14 questions below, each with the working. Every answer was checked by a second pass before it was published.
Volume formulas and their justification, cross-sections, rotations of two-dimensional objects, modelling with solids, density.
How this unit is tested
Start by locking down the four volume formulas — prism/cylinder, pyramid/cone, sphere — and be able to explain WHY each one is true, not just recite it. The justification questions on the exam usually ask you to reason from Cavalieri's Principle (equal cross-sections at every height means equal volume) or from a decomposition argument (a prism splits into three congruent pyramids). Practice explaining these in a sentence or two before you ever plug in numbers.
For cross-sections, always ask two things: what shape am I cutting, and is the plane parallel, perpendicular, or at an angle to the solid's axis or base? The same solid gives different cross-sections depending on the cut — a cylinder cut perpendicular to its axis gives a circle, parallel to its axis gives a rectangle, and at an angle gives an ellipse. Sketch the cut before you name the shape.
For rotations, identify the 2D figure, the axis it spins around, and which dimension becomes the radius versus the height of the resulting solid. A common trap is rotating about a different side than the problem states, which swaps radius and height. Draw the axis as a vertical or horizontal line first.
For modeling and density problems, break the real object into simple solids (cylinders, cones, spheres, prisms), find each volume, then combine (add for a solid built from parts, subtract for a hole or cavity). Density is just mass divided by volume — the geometry is in getting the volume right; the arithmetic is in keeping units consistent throughout.
What you have to know
Cavalieri's Principle
If two solids have the same height and every cross-section parallel to their bases at the same distance from the base has equal area, then the two solids have equal volume.
Volume of a prism or cylinder
$V = Bh$, where $B$ is the area of the base and $h$ is the height (perpendicular distance between the bases).
Volume of a pyramid or cone
$V = \frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the height.
Volume of a sphere
$V = \frac{4}{3}\pi r^{3}$, where $r$ is the radius.
Density formula
Density $= \dfrac{\text{mass}}{\text{volume}}$, so mass $=$ density $\times$ volume when volume is known.
14 practice questions
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An oblique cylinder and a right cylinder both have base radius 5 cm and height 12 cm. What is the volume of each, and why must they be equal?
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Answer. Both have volume $300\pi$ cubic centimeters, because Cavalieri's Principle says equal-height solids with equal cross-sectional area at every level have equal volume.
Volume of a cylinder is $\pi r^{2}h = \pi(5)^{2}(12) = 300\pi$. Since both cylinders have the same circular base area at every height (just shifted sideways in the oblique case), Cavalieri's Principle guarantees equal volumes even though one is slanted. -
A right circular cone has radius 6 and height 9. Find its exact volume.
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Answer. $108\pi$ cubic units.
Use $V=\frac{1}{3}\pi r^{2}h = \frac{1}{3}\pi(6)^{2}(9) = \frac{1}{3}\pi(36)(9) = 108\pi$. The 1/3 factor is essential because a cone comes to a point rather than having two congruent bases. -
A plane parallel to the base slices through a right circular cone. What shape is the resulting cross-section?
- A triangle
- A smaller circle
- An ellipse
- A trapezoid
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Answer. A circle, smaller than the base circle.
Cutting a cone with a plane parallel to its circular base always produces a smaller circle, similar to the base, because every horizontal slice of a cone is a scaled-down copy of the base. -
A plane slices through a right circular cylinder at an angle to its axis, neither parallel nor perpendicular to it. What shape is the cross-section?
- A circle
- A rectangle
- An ellipse
- A parabola
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Answer. An ellipse.
A perpendicular cut gives a circle and a parallel cut gives a rectangle, but a slanted cut through the curved surface stretches the circle into an ellipse. -
A sphere has radius 10. A plane cuts the sphere at a perpendicular distance of 6 from the center. Find the area of the circular cross-section.
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Answer. $64\pi$ square units.
The cross-section radius satisfies $r_{cs}^{2}+d^{2}=R^{2}$, so $r_{cs}^{2}=10^{2}-6^{2}=64$, giving $r_{cs}=8$. Area is $\pi r_{cs}^{2}=64\pi$. -
A rectangle with sides 4 and 9 is rotated 360° about the side of length 9. Describe the resulting solid and find its exact volume.
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Answer. A cylinder with radius 4 and height 9, volume $144\pi$.
The side of length 9 stays fixed as the axis (becomes the height), while the side of length 4 sweeps out the circular base (becomes the radius). $V=\pi r^{2}h=\pi(4)^{2}(9)=144\pi$. -
A right triangle with legs 3 and 4 is rotated 360° about the leg of length 4. Find the exact volume of the resulting solid.
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Answer. $12\pi$ cubic units.
Rotating about the leg of length 4 makes that leg the height and the leg of length 3 the radius, producing a cone. $V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi(3)^{2}(4)=12\pi$. -
A semicircle of radius 5 is rotated 360° about its diameter. Find the exact volume of the resulting solid.
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Answer. $\frac{500}{3}\pi$ cubic units.
Rotating a semicircle about its diameter produces a full sphere of the same radius. $V=\frac{4}{3}\pi r^{3}=\frac{4}{3}\pi(5)^{3}=\frac{500}{3}\pi$. -
A rectangular prism has volume V. It is decomposed into three pyramids of equal volume, all sharing the same base as the prism and the same height as the prism. What is the volume of one of these pyramids in terms of V, and why does this justify the pyramid volume formula?
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Answer. Each pyramid has volume $\frac{V}{3}$, which shows $V_{pyramid}=\frac{1}{3}Bh$ since the prism's volume is $Bh$.
Since the prism splits exactly into three congruent pyramids with the same base and height, each pyramid must have one third of the prism's volume. Because the prism's volume is $Bh$, each pyramid's volume is $\frac{1}{3}Bh$, which is exactly the pyramid volume formula. -
A silo is modeled as a cylinder of radius 5 ft and height 20 ft topped by a cone of height 6 ft with the same radius. Find the total exact volume of the silo.
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Answer. $550\pi$ cubic feet.
Cylinder volume: $\pi(5)^{2}(20)=500\pi$. Cone volume: $\frac{1}{3}\pi(5)^{2}(6)=50\pi$. Total: $500\pi+50\pi=550\pi$ cubic feet, since the composite solid's volume is the sum of its parts. -
A cylindrical pipe has an outer radius of 6 cm, an inner radius of 4 cm (a hollow center running through it), and length 30 cm. Find the exact volume of the material the pipe is made of.
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Answer. $600\pi$ cubic centimeters.
Find the outer cylinder's volume and subtract the inner (hollow) cylinder's volume: $\pi(6)^{2}(30)-\pi(4)^{2}(30) = 1080\pi - 480\pi = 600\pi$. Subtracting removes the hole from the solid material. -
A square pyramid has a base with side length 8 m and height 12 m. A plane parallel to the base cuts the pyramid at a height of 4 m above the base. Find the area of the cross-section formed by this cut.
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Answer. $\frac{256}{9}$ square meters, approximately 28.4 square meters.
The cross-section is a smaller square similar to the base, scaled by the ratio of remaining height to total height from the apex: the cut is 4 m up out of 12 m, so the distance from apex is 8 m out of 12 m, giving a scale factor of $\frac{8}{12}=\frac{2}{3}$. The side length becomes $8\times\frac{2}{3}=\frac{16}{3}$, so the area is $\left(\frac{16}{3}\right)^{2}=\frac{256}{9}$. -
A solid metal sphere has radius 3 cm and a measured mass of 452 grams. Find its density, rounded to the nearest hundredth of a gram per cubic centimeter.
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Answer. Approximately 4.00 grams per cubic centimeter.
Volume of the sphere: $V=\frac{4}{3}\pi(3)^{3}=36\pi\approx113.1$ cubic centimeters. Density = mass/volume = $452/113.1\approx4.00$ grams per cubic centimeter. -
A rectangular block of wood measures 2 cm by 3 cm by 10 cm and has a mass of 84 grams. Find its density, and determine whether it would float in water, given that water has a density of 1 gram per cubic centimeter.
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Answer. Density is 1.4 grams per cubic centimeter, so the block would sink because it is denser than water.
Volume = $2\times3\times10=60$ cubic centimeters. Density = $84/60=1.4$ grams per cubic centimeter. Since 1.4 is greater than water's density of 1 gram per cubic centimeter, the block sinks.
What people get wrong
- Forgetting the factor of 1/3 for cones and pyramids — students often compute $Bh$ instead of $\frac{1}{3}Bh$. Always write the formula down first and check whether the solid comes to a point (cone/pyramid, needs the 1/3) or has two parallel congruent bases (prism/cylinder, does not).
- Rotating about the wrong axis, which swaps which side becomes the radius and which becomes the height. Before computing, draw the axis of rotation as a line and label which dimension of the 2D shape lies along it and which sticks out perpendicular to it.
- Assuming a plane cutting a cylinder or cone at any angle always gives the same shape as a perpendicular cut. A plane through a cylinder at an angle to the axis gives an ellipse, not a circle, and a plane through a cone at an angle can give a circle, ellipse, parabola, or hyperbola depending on the angle — for this course, focus on parallel-to-base (circle) versus through-the-axis (triangle) cuts.
- Treating an oblique prism or cylinder as needing a different volume formula. By Cavalieri's Principle, an oblique solid has the same volume as a right solid with the same base area and the same perpendicular height — the slant does not change the volume.
- Mixing units when computing density or mass, such as leaving one measurement in centimeters and another in meters. Convert every length to the same unit before computing volume, and match the volume's unit to the density's unit before multiplying.
Drill this unit until it sticks
These questions come back on a schedule built from what you get wrong, alongside the rest of Geometry. Free, and no account needed to start.