Solid Geometry: Chapter Review

Vocabulary and notation

Polyhedron

A solid bounded entirely by polygonal faces. The straight segments shared by faces are edges, and the corners are vertices; a circular rim is not a polyhedron edge.

Prism and pyramid

A prism joins two congruent parallel polygonal bases by parallel edges. A pyramid joins one polygonal base to an off-plane apex by triangular faces. Circular counterparts are the cylinder and cone.

Right and oblique

A right prism’s joining edges and a right cylinder’s axis are perpendicular to their base planes. A right cone’s apex is directly over its base center; obliqueness concerns the solid, regardless of page orientation.

Parallel and skew lines

Distinct parallel lines are coplanar and disjoint. Skew lines are noncoplanar, so they neither meet nor run parallel.

Lines and planes

Distinct planes are parallel or intersect in a complete line. A line parallel to a plane has no point in it; a perpendicular line meets every plane line through its foot at a right angle.

Net

A connected flat arrangement of surface pieces that folds into the complete boundary without gaps or overlapping interiors. Cut edges can meet again; separated faces can become neighbors.

Orthographic view

A projection along parallel sight lines. Front records width and height; top, width and depth; right, depth and height.

Cross-section and axis of rotation

A cross-section is the intersection with a plane, with no thickness. An axis of rotation is the fixed line about which a filled plane region turns to sweep out a solid.

Base area $B$, perimeter $P$, height $h$

B and P measure one base. Height h is perpendicular separation between base planes or from an apex to its base plane.

Slant height $\ell$ and apothem $a$

A regular pyramid’s slant height is its common triangular face altitude; its base apothem joins the center to a side midpoint. A right cone’s slant height joins its apex to the base rim.

Lateral and total surface area

Lateral area excludes bases. Total area counts the stated boundary, including any exposed rear or cavity surfaces.

Sphere, ball, great circle

A sphere is the surface at a fixed positive distance from its center; a ball is that surface and its interior. A central cut gives a great circle with the sphere radius.

Annulus

The ring between concentric circles. A thick open pipe has annular ends, an outer wall, and an inner bore wall.

Similar solids

Uniformly resized copies, allowing a move, turn, or reflection. Every corresponding length uses one positive factor.

Formulas and theorems

• Prism and pyramid counts

  $$\begin{gathered} \text{prism: }(n+2,\,3n,\,2n)\\ \text{pyramid: }(n+1,\,2n,\,n+1) \end{gathered}$$

  Use when Counts are faces, edges, and vertices, in that order. The polygonal base has $n\ge3$ sides. Include hidden parts; changing a prism’s lean does not change the counts.

• True section shapes

  Base-parallel sections repeat a prism or cylinder’s base, or shrink toward a pyramid or cone’s apex. A right cylinder’s axial section is rectangular; a right cone’s is isosceles triangular.

  Use when Interior cuts of positive area. An oblique section is elliptical if its entire closed boundary lies on the curved side of a right circular cylinder or cone, missing flat ends and any apex.

• Cavalieri’s principle

  Two ordinary solids between the same parallel boundary planes have equal volumes if every parallel cutting plane gives equal section areas.

  Use when Align both solids over the same perpendicular height and compare every corresponding level. Section shapes may differ; matching only the base or several sampled levels is insufficient.

• Prism and cylinder volume

  $$\begin{gathered} V_{\text{prism}}=Bh\\ V_{\text{cylinder}}=\pi r^2h \end{gathered}$$

  Use when Positive base area and perpendicular height; positive radius for a circular cylinder. Applies to right and oblique cases.

• Right prism boundary

  $$\begin{gathered} L=Ph\\S_{\text{closed}}=Ph+2B \end{gathered}$$

  Use when Right prism only: its lateral rectangles share height h. Include only the bases present. For an oblique prism, use actual lateral face areas; their altitudes may differ from h.

• Right cylinder boundary

  $$\begin{gathered} L=2\pi rh\\S_{\text{closed}}=2\pi rh+2\pi r^2 \end{gathered}$$

  Use when Right circular cylinder with positive radius and height. Its unrolled side is a rectangle of width equal to the circumference. Include zero, one, or two end disks as specified; thickness can add surfaces.

• Pyramid and cone volume

  $$\begin{gathered} V_{\text{pyramid}}=\tfrac13Bh\\V_{\text{cone}}=\tfrac13\pi r^2h \end{gathered}$$

  Use when Positive perpendicular height; a simple polygonal base of positive area or a circular base of positive radius. Applies to right and oblique cases, including a height foot outside the base.

• Parallel sections of pointed solids

  $$\begin{gathered} \text{length factor}=\frac th\\A(t)=B\left(\frac th\right)^2 \end{gathered}$$

  Use when Pyramid or circular cone, with $B,h>0$. Measure perpendicular distance t from the apex toward a base-parallel section, $0\le t\le h$. For a level z above the base, use $t=h-z$.

• Perpendicular height and slant height

  $$\begin{gathered} \text{right cone: }\ell^2=h^2+r^2\\\text{regular pyramid: }\ell^2=h^2+a^2 \end{gathered}$$

  Text description

  Half of a right cone’s axial section is a right triangle with apex V above base center O, and rim point M to the right. VO is the perpendicular height h, OM is the base radius r, and VM is the slant height ell. The right angle is at O. Slant height is the hypotenuse, separate from the perpendicular height.

  Use when Positive dimensions. In the pyramid, a reaches a side midpoint, not a vertex; for square side s, $a=s/2$. These right triangles require the stated right or regular solid.

• Regular pyramid surface

  $$\begin{gathered} L=\tfrac12P\ell\\S_{\text{closed}}=\tfrac12P\ell+B \end{gathered}$$

  Use when Convex regular base and apex directly above its center; all side faces share slant height $\ell$. If face altitudes differ, add their individual triangle areas instead. Include the base only when present.

• Right cone surface and sector

  $$\begin{gathered} L=\pi r\ell\\S_{\text{closed}}=\pi r\ell+\pi r^2\\\text{sector radius}=\ell\\\text{sector arc}=2\pi r\\\theta=360^\circ\frac r\ell \end{gathered}$$

  Use when Right circular cone, $r,h>0$, so $\ell>r$ and $0<\theta<360^\circ$. The whole selected arc becomes the rim without overlap. An open base adds no disk; an oblique cone has no single-slant-height shortcut.

• Sphere measures and sections

  $$\begin{gathered} S=4\pi R^2\\V_{\text{ball}}=\tfrac43\pi R^3\\\rho^2=R^2-d^2 \end{gathered}$$

  Use when Sphere radius $R>0$. A plane at perpendicular center distance $0\le d<R$ cuts a circle of radius $\rho$; the ball section is its disk. At $d=R$ it touches one point; at $d>R$ it misses.

• Hemisphere measures

  $$\begin{gathered} V=\tfrac23\pi r^3\\S_{\text{curved}}=2\pi r^2\\S_{\text{total}}=3\pi r^2 \end{gathered}$$

  Text description

  The curved cap of a hemisphere rises above its circular flat base disk, shown as an ellipse in perspective. A label above identifies the curved cap; a leader below identifies the flat disk. The curved cap and flat disk are separate boundary pieces; total hemisphere area includes both.

  Use when Positive radius and a cut through the sphere center. Total includes one flat disk; curved area excludes it. A support covering that disk leaves only the uncovered cap exposed.

• Exterior area at a join

  $$S_{\text{joined}}=S_1+S_2-2A$$

  Use when Two solids with nonoverlapping interiors share exactly one matching patch of area A. Start with both separate total surface areas; no other contacts or excluded surfaces.

• Open right pipe

  $$\begin{gathered} V=\pi(R^2-r^2)h\\L_{\text{outer}}=2\pi Rh\\L_{\text{inner}}=2\pi rh\\A_{\text{ring}}=\pi(R^2-r^2)\\S=L_{\text{outer}}+L_{\text{inner}}+2A_{\text{ring}} \end{gathered}$$

  Text description

  Two true sections of the same open right circular pipe are shown schematically. The end section is a ring of material around an empty circular bore. In the axial section, two rectangular strips of material border an empty passage which remains open at both ends. The inner bore wall and the two annular end rings are material surfaces as well as the outside wall.

  Use when Outer radius R, inner r, length h: $0<r<R$ and $h>0$. A coaxial bore runs through the full length. V measures material. Include both walls and both end annuli for total exposed area; thickness is $R-r$.

• Rectangular box diagonal

  $$\begin{gathered} d=\sqrt{a^2+b^2+c^2} \end{gathered}$$

  Use when Positive perpendicular edge lengths a, b, c. First find a face diagonal, then combine it with the perpendicular third edge. The formula does not apply to an oblique box.

• Uniform similarity powers

  $$\begin{gathered} k=\frac{\text{image length}}{\text{original length}}>0\\\frac{S_i}{S_o}=k^2\\\frac{V_i}{V_o}=k^3\\\frac{S_i/V_i}{S_o/V_o}=\frac1k \end{gathered}$$

  Use when Similar solids with positive areas and volumes; i is image and o is original. Use consistent units and corresponding boundary pieces, including openings and joins. Surface area per volume has inverse-length units.

• Recover a length scale

  $$\begin{gathered} k=\sqrt{\frac{S_i}{S_o}}\\k=\sqrt[3]{\frac{V_i}{V_o}} \end{gathered}$$

  Use when Similarity must already be established; use positive areas, volumes, and roots, with matching measured boundaries and units. A volume or area ratio alone does not prove similarity.

  e.g. For similar solids with volume ratio image:original $512:2197$, find the length factor: $8/13$.

Problem types, step by step

Exam traps

• Trap Trusting a perspective angle or apparent oval

  Fix Use stated spatial relationships and the cutting plane. A drawing can shorten lengths and show a true circle as an oval.

• Trap Calling every pair of disjoint lines parallel

  Fix Check complete lines and coplanarity. Noncoplanar lines are skew; two edge segments not touching is insufficient evidence.

• Trap Using a slanted length as volume height

  Fix Volume uses perpendicular separation. For pyramid and cone lateral area, use the face altitude or common slant height required by the formula.

• Trap Using equal bases and heights for arbitrary solids

  Fix Cavalieri needs equal section areas at every corresponding height. Finite samples do not supply this requirement.

• Trap Squaring the wrong radius or section fraction

  Fix Halve a diameter first. For pointed parallel sections, measure from the apex and square the length fraction for area.

• Trap Halving sphere area for a whole hemisphere boundary

  Fix Half gives the curved cap only. Add one flat disk for total area; use half-ball volume for enclosed space.

• Trap Turning the removed cone upside down in the hemisphere comparison

  Fix Place its apex at the cylinder’s bottom center and its base across the top. Cylinder radius and height are both R. At height z above the bottom, the removed radius is z; both retained sections have area $\pi(R^2-z^2)$.

• Trap Treating the radial sphere-area sketch as exact finite pyramids

  Fix Curved patches are replaced by flat approximations, whose heights approach the radius under refinement. The argument is informal and limiting.

• Trap Subtracting an inner surface because volume was removed

  Fix Subtract removed volume but add newly exposed walls. A through bore removes two end patches; a blind cavity removes one and exposes a floor.

• Trap Scaling only the visible dimensions or mismatching boundaries

  Fix Similarity powers require the entire solid and corresponding measured surfaces to scale together. A one-axis stretch needs direct formulas.

• Trap Mixing length, area, volume, and capacity units

  Fix Use one length unit before calculating: area is squared, volume cubed, and $S/V$ has inverse-length units. $1\text{ L}=1000\text{ cm}^3$ and $1\text{ mL}=1\text{ cm}^3$; capacity uses inside dimensions.