3.5 3D Figures and Formulas

Why This Section Matters

Three-dimensional figures close the geometry domain on Praxis 5165. You must compute surface area and volume for prisms, cylinders, cones, pyramids, and spheres, and interpret cross sections of solids. Formula items are straightforward if you know which measurement is asked — total surface area versus lateral area trips many candidates.

Memorize the high-yield formulas below; ETS provides an on-screen calculator for arithmetic but not for recalling structure. Three-dimensional items often appear as word problems with a diagram — label radius, height, and slant height before substituting.

Reference Table: Volume and Surface Area

Cavalieri's principle: cross-sections at equal heights have equal area ⇒ solids have equal volume. A slanted cylinder with the same base and height as a right cylinder has the same volume.

Prisms and Cylinders

A prism has two congruent parallel bases; volume equals base area times height (B · h). A cylinder is a circular prism: V = πr²h.

Lateral surface area of a cylinder is 2πrh (the label on a can, excluding top and bottom). Total surface area adds the two circular bases.

Worked Example: Cylinder Volume

Radius 3, height 5:

V = π(3²)(5) = π(9)(5) = 45π cubic units.

Leave answers in terms of π unless the stem requests a decimal.

Worked Example: Cylinder Surface Area

Same cylinder (r = 3, h = 5):

• Two bases: 2πr² = 2π(9) = 18π
• Lateral area: 2πrh = 2π(3)(5) = 30π
• Total SA = 18π + 30π = 48π square units

Worked Example: Rectangular Prism

A box measures 4 cm by 5 cm by 6 cm.

V = 4 · 5 · 6 = 120 cm³.

SA = 2(20 + 24 + 30) = 2(74) = 148 cm².

Cones and Pyramids

Both cones and pyramids include the factor 1/3 before base area times height. If a cone and cylinder share the same base radius and height, the cone's volume is exactly one-third of the cylinder's.

For a cone, slant height ℓ, radius r, and height h satisfy ℓ² = r² + h² by the Pythagorean theorem.

Worked Example: Cone Volume

Radius 6, height 9:

V = (1/3)π(6²)(9) = (1/3)π(36)(9) = (1/3)(324π) = 108π cubic units.

Worked Example: Slant Height

Cone with r = 5 and h = 12. Find ℓ.

ℓ = √(5² + 12²) = √(25 + 144) = √169 = 13.

Spheres

Sphere formulas appear less often than cylinders but still show up:

• Volume V = (4/3)πr³
• Surface area SA = 4πr²

Worked Example: Sphere Volume

Radius 3:

V = (4/3)π(27) = 36π cubic units.

Worked Example: Hemisphere

A solid hemisphere (half-sphere) of radius 4 includes a flat circular base.

Volume = (1/2)(4/3)π(64) = (128π/3) cubic units.

Curved surface area = (1/2)(4πr²) = 2πr² = 32π (plus πr² for the flat base if total exterior area is requested).

Cross Sections and Nets

A horizontal cross section of a right cylinder is a circle; of a rectangular prism, a rectangle matching the base shape. Slicing a sphere produces a circle (or a point at a tangent cut).

A net unfolds faces into a plane. For a cube, the net has six squares; total surface area equals the sum of their areas.

Teaching items may ask which cross section is possible when slicing a solid — eliminate shapes that cannot arise from a single plane cut.

Composite Solids

Break composite figures into familiar pieces. Example: a silo with a cylindrical body (r = 4, h = 20) and hemispherical top (same r):

• Cylinder volume: π(16)(20) = 320π
• Hemisphere volume: (1/2)(4/3)π(64) = (128π/3)
• Total ≈ combine with common denominator when needed

Watch units: if radius is in centimeters and height in meters, convert before multiplying.

Common Traps

• Using 2πrh alone when the item asks for total surface area (missing the two bases).
• Forgetting 1/3 on cone or pyramid volume.
• Squaring diameter instead of radius (r = d/2).
• Mixing slant height ℓ with height h in cone formulas.
• Adding volumes of composites without matching units.

Section Checklist

• Write the formula, identify r, h, ℓ, and B, then substitute.
• State whether the item wants lateral area or total surface area.
• For composites, sketch the decomposition before calculating.
• Use ℓ² = r² + h² when a cone problem gives slant height indirectly.

Density and Capacity Applications

Applied items may give volume in cubic centimeters and ask for mass using density: mass = density × volume. Others convert liters to cubic centimeters (1 L = 1000 cm³). Set up the geometry first, then apply the unit conversion as a second step so radius-height errors do not compound.

Scaling Solids

If every dimension of a solid is scaled by factor k, volume scales by k³. Doubling edge length multiplies volume by 8. This parallels the k² area scaling rule for similar plane figures in Section 3.1.

Link to Praxis Practice

Cylinder volume and cone-to-cylinder comparison items are common in the 5165 bank. Memorize V = πr²h and the one-third cone factor as a pair so you can move quickly between related stems without re-deriving each time.