Study guide
11.1 Volume of Cylinders, Cones, and Spheres
ACCUPLACER Advanced Algebra and Functions Exam
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1 Introduction & Exam Overview
1.1 Current Exam Facts 1.2 Scoring, Skills Insight Bands, and Placement 1.3 How to Study for AAF
2 Linear Equations
2.1 Writing Linear Equations in One and Two Variables 2.2 Solving Linear Equations 2.3 Linear Inequalities 2.4 Systems of Two Linear Equations
3 Linear Applications and Graphs
3.1 Slope, Intercepts, and Graphing Lines 3.2 Parallel and Perpendicular Lines 3.3 Modeling with Linear Functions 3.4 Graphing Linear Inequalities and Systems
4 Factoring
4.1 GCF, Grouping, and Special Products 4.2 Factoring Quadratic Trinomials 4.3 Factoring Cubics and Higher Polynomials
5 Quadratics
5.1 Quadratic Forms, Graphs, Vertex, and Intercepts 5.2 Solving Quadratics by Factoring 5.3 Quadratic Formula and Completing the Square 5.4 Quadratic Inequalities and Quadratic-Linear Systems
6 Functions
6.1 Function Notation, Evaluation, Domain, and Range 6.2 Graphs of Functions, Vertical Line Test, Maxima and Minima 6.3 Function Operations, Composition, and Inverses 6.4 Transformations and Interpreting Functions in Context
7 Radical and Rational Equations
7.1 Simplifying Radical Expressions 7.2 Solving Radical Equations and Extraneous Roots 7.3 Rational Expressions and Equations 7.4 Graphs, Domain, and Range of Radical and Rational Functions
8 Polynomial Equations
8.1 Polynomial Operations and Structure 8.2 Solving Polynomial Equations 8.3 Graphing Polynomials, Roots, and End Behavior
9 Exponential and Logarithmic Equations
9.1 Exponential Functions, Growth, and Decay 9.2 Graphs of Exponential Functions 9.3 Logarithms and Logarithmic Properties 9.4 Solving Exponential and Logarithmic Equations
10 Geometry Concepts for Algebra 1
10.1 Area, Perimeter, and Prism Volume 10.2 Distance Formula and Pythagorean Theorem 10.3 Dilations, Rotations, Translations, and Reflections
11 Geometry Concepts for Algebra 2
11.1 Volume of Cylinders, Cones, and Spheres 11.2 Intersecting Lines and Angle Relationships 11.3 Triangle Similarity and Congruence 11.4 Circles in the Coordinate Plane
12 Trigonometry
12.1 Right-Triangle Trigonometry and Special Triangles 12.2 Unit Circle, Radians, and Arc Length 12.3 Trigonometric Graphs and Identities 12.4 Solving Trig Equations; Law of Sines and Cosines
11.1 Volume of Cylinders, Cones, and Spheres
Key Takeaways Explain This Topic AI
AAF weights Geometry concepts for Algebra 2 at 5–10% of the 20-item CAT, typically 1–2 questions; Table 11 starts that block with volume of nonprism objects.
Cylinder V = πr^2 h, cone V = (1/3)πr^2 h, sphere V = (4/3)πr^3. A cone is one-third of the cylinder that shares its base and height.
A cone with radius 4 and height 9 has volume 48π; 144π is the cylinder with the same r and h and is the standard forgotten-1/3 trap.
AAF often leaves volume in terms of π. Convert diameter to radius before squaring or cubing, and never use slant height as perpendicular height.
Prism volume V = Bh from Algebra 1 geometry still describes a cylinder when B = πr^2; cones and spheres are not prisms and do not use a bare Bh.
Last updated: August 2026
11.1 Volume of Cylinders, Cones, and Spheres
College Board’s Next-Generation ACCUPLACER Advanced Algebra and Functions (AAF) test weights Geometry concepts for Algebra 2 at 5–10% of the 20-item computer-adaptive exam — typically 1–2 items . Table 11 names four Algebra 2 geometry skills: volume of nonprism objects, intersecting line theorems, triangle similarity and congruency, and circle equations in the coordinate plane. This section is the first of those four: cylinders, cones, and spheres . Prism volume — rectangular, triangular, and other polygonal bases — already lives in Area, Perimeter, and Prism Volume . AAF splits the circular solids into Algebra 2 because the formulas change.
A cylinder is the circular analogue of a prism, so its formula still looks like V = Bh . A cone and a sphere are not prisms. The CAT will not hand you a formula sheet. If you copy the prism rule onto a cone, you lose the 1/3 factor — the highest-frequency volume trap on this slice of the exam. Work FREE mixed AAF items at /practice/accuplacer-advanced-algebra after you can produce exact answers in terms of π and recognize when a stem wants a decimal. Official category names live on College Board’s What’s on the Tests page; that student page currently lists nine named skill categories and omits geometry , so study from Table 10/11, not from the abbreviated public list.
The three formulas
Solid Volume What B and h mean Prism cousin
Cylinder V = πr^2 h B = πr^2 is the circular base; h is perpendicular height Same structure as V = Bh for a prism
Cone V = (1/3)πr^2 h Same B and h as the cylinder that shares the base and height One-third of that cylinder — not a prism
Sphere V = (4/3)πr^3 No base and no height; only the radius No prism analogue
A cylinder has two parallel congruent circular bases and a lateral surface. Height is the perpendicular distance between the bases, not a slanted generator. A right cylinder — the AAF default unless a stem says otherwise — has the side perpendicular to the bases. An oblique cylinder still uses perpendicular height, but AAF almost always draws a right cylinder.
A cone has one circular base and an apex. Height is the perpendicular distance from the apex to the plane of the base. Slant height is a different length, used for lateral surface area, not for volume. If the stem gives a slant height ℓ and a radius, recover h with Pythagoras before you plug into volume: ℓ^2 = r^2 + h^2 .
A sphere is the set of points in space at a fixed distance r from a center. Volume uses r^3 , not r^2 . Surface area 4πr^2 is the common mix-up: if an option looks like a surface-area number on a volume question, it is a distractor.
When AAF leaves π in the answer
AAF is multiple-choice. Many volume stems have options such as 48π , 144π , and 192 . Leave the answer in terms of π unless the stem asks for a nearest integer, a decimal, or a calculator-enabled numeric value. Handheld calculators are not allowed except with an approved accommodation (College Board calculator policy). Some items show an on-screen calculator icon in the top-right corner; even then, exact kπ form is often among the choices and is faster than multiplying by 3.14.
Do not replace π with 3.14 unless the stem tells you to approximate. 300π and 942 are different answers; if both appear, the exact form is the one that matches πr^2 h without rounding.
- Cylinder: worked numeric examples
- Worked: radius 5 cm, height 12 cm
V = πr^2 h = π(5)^2(12) = π(25)(12) = 300π cubic centimeters.
If the stem wanted a decimal with π ≈ 3.14, you would compute 300 × 3.14 = 942 . Do not write that unless asked. A trap option 60π multiplies r · h instead of r^2 · h . A trap 150π halves the height or the squared radius. Another trap uses diameter 10 in place of radius: π(10)^2(12) = 1200π .
Worked: diameter 8 in, height 7 in
The radius is 4 , not 8. V = π(4)^2(7) = 16π · 7 = 112π cubic inches.
Reading “diameter 8” and plugging r = 8 is the classic cylinder error. Always convert diameter to radius first: r = d/2 .
A cylinder is the circular version of the prism formula from Algebra 1 geometry. For a rectangular prism, B is length times width. For a triangular prism, B is (1/2)bh of the triangular base. For a cylinder, B is πr^2 . Same skeleton, different base. That is why Table 11 splits “expressions for area, perimeter, and volume” (Algebra 1, including prisms) from “volume of nonprism objects” (Algebra 2, including the cone and sphere, with the cylinder riding along as the circular Bh case).
Cone: the 1/3 factor
A cone with the same circular base and the same perpendicular height as a cylinder has exactly one-third of that cylinder’s volume. The formula is not a decoration; it is the relationship.
- V_cone = (1/3) V_cylinder = (1/3)πr^2 h
- Worked: radius 4 in, height 9 in
V = (1/3)π(4)^2(9) = (1/3)π(16)(9) = (1/3)(144π) = 48π cubic inches.
Trap: forgetting the 1/3 and reporting 144π . That is the cylinder with the same r and h, and it will almost always sit in the option list. Trap: writing (1/2)πr^2 h by borrowing the triangle-area half. A cone is not a triangular area of revolution in that sense; the factor is one-third , the same factor that turns any pyramid into one-third of the prism with the same base and height. Trap: using slant height 9 as if it were perpendicular height.
Worked: recover height from slant height
A right cone has radius 5 cm and slant height 13 cm. The perpendicular height is not 13.
h = √(ℓ^2 − r^2) = √(169 − 25) = √144 = 12
Then V = (1/3)π(25)(12) = 100π cubic centimeters.
If you plug ℓ = 13 into the volume formula you get (1/3)π(25)(13) = 325π/3 , which is wrong. Volume never uses slant height.
Worked: cone compared with its cylinder
A can is a cylinder of radius 3 cm and height 10 cm. A conical paper cup has the same radius and height.
- Cylinder: V = π(9)(10) = 90π
- Cone: V = 30π
The cup holds one-third of the can. An AAF stem can give you the cylinder volume and ask for the cone, or give the cone and ask for the cylinder. Multiply or divide by 3; do not re-derive πr^2 unless you need a check.
- Sphere: r cubed, not r squared
- V = (4/3)πr^3
Cube the radius first, then multiply by 4/3 and by π. Order matters for mental arithmetic: (4/3)π(r^3) , not ((4/3)πr)^3 .
- Worked: radius 6 m
- r^3 = 216
V = (4/3)π(216) = (4 · 216 / 3)π = (864/3)π = 288π cubic meters.
Trap: (4/3)π(36) = 48π uses r^2 (the surface-area cousin) instead of r^3 . Trap: (4/3)π(6) = 8π forgets to raise r at all. Trap: 4π(216) = 864π drops the denominator 3. Trap: (4/3)π(18) = 24π cubes incorrectly ( 6^3 is 216, not 18).
Worked: diameter 10 ft
Radius is 5. r^3 = 125 . V = (4/3)π(125) = 500π/3 cubic feet.
Leave the answer as 500π/3 if that matches an option. Do not force a mixed number or a decimal. If options are 500π/3 , 2000π/3 (used diameter as radius), 100π (used (4/3)πr^2 ), and 250π/3 (halved after cubing), pick 500π/3 .
Hemisphere
A hemisphere is half a sphere: V = (2/3)πr^3 . AAF composite items often glue a hemisphere onto a cylinder (a tank with a rounded cap) or onto a cone (an ice-cream scoop on a wafer). Compute each piece, then add. Do not apply one formula to the whole picture.
- Composite solids
- Worked: silo = cylinder + hemisphere
A grain silo is a cylinder of radius 4 m and height 9 m, capped by a hemisphere of the same radius (the flat face of the hemisphere is the top of the cylinder, so you do not double-count a base).
- Cylinder: π(16)(9) = 144π
- Hemisphere: (2/3)π(64) = 128π/3
Total: 144π + 128π/3 = 432π/3 + 128π/3 = 560π/3 cubic meters.
If the cap were a cone of height 3 m instead of a hemisphere:
- Cone: (1/3)π(16)(3) = 16π
- Total: 144π + 16π = 160π
Sketch which pieces are present before you reach for a formula. AAF distractors compute only the cylinder, only the cap, or the cap with the wrong 1/2 vs 1/3 vs 2/3 factor.
Prism contrast (Algebra 1 vs Algebra 2)
From Area, Perimeter, and Prism Volume : a prism has two parallel polygonal bases and rectangular (or parallelogram) lateral faces. V = Bh with B the area of one polygonal base.
- Object Base Volume
- Rectangular prism rectangle ℓw ℓwh
- Triangular prism triangle (1/2)bh (1/2)bh · H
- Cylinder circle πr^2 πr^2 h
- Square pyramid square s^2 (1/3)s^2 h
- Cone circle πr^2 (1/3)πr^2 h
- Sphere none (4/3)πr^3
Pyramids and cones share the 1/3 . Prisms and cylinders share no extra fraction . Spheres are their own family. If a stem shows a triangular prism, you are still in Algebra 1 geometry. If it shows a cone or a sphere, you are in this section. A cylinder can appear in either narrative, but the formula is the same πr^2 h .
AAF volume checklist
Name the solid. Cylinder, cone, sphere, hemisphere, or a composite of those.
Convert diameter to radius before you square or cube.
For a cone, write the 1/3 before you multiply. For a sphere, write 4/3 and cube r .
Height for volume is perpendicular height, never slant height. Recover h from ℓ and r if needed.
Leave π in the product unless the stem asks you to approximate.
Units are cubic. An option with πr^2 only is an area, not a volume.
For composites, compute each piece and add. Do not invent a single mega-formula.
Geometry is only 1–2 CAT items in the Algebra 2 geometry block, so a missed 1/3 on a cone is an expensive miss. The next section, Intersecting Lines and Angle Relationships , is the angle-relationships partner to this volume skill.