12. GED® Strategy: Pick the Solid, Pick the Formula

Session 12 Learning Plan

Learning objective: Select, combine, estimate, and verify volume and surface-area strategies under GED®-style conditions.

Key vocabulary: formula sheet, composite solid, internal face, estimation, error analysis

Success criterion: Solve mixed problems with a documented target-to-formula routine and a complete final error audit.

Learn and Practice

The GED® gives you a formula sheet, so your job is to choose the right formula and put the numbers in correctly. Use this routine:

• What solid is it? Box, cube, cylinder, cone, or sphere.
• Volume or surface area? Fill/hold = volume (cubic units); cover/wrap = surface area (square units).
• Radius or diameter? Circles need the radius; halve the diameter if needed.
• Square or cube the radius? Cylinder and cone area use \(r^2\); a sphere's volume uses \(r^3\).
• Any fraction out front? Cone has \(\tfrac{1}{3}\); sphere has \(\tfrac{4}{3}\).

Volume quick-reference:

• Box: \(V = lwh\)
• Cube: \(V = s^3\)
• Prism / cylinder: \(V = Bh\) (base area times height); cylinder \(= \pi r^2 h\)
• Cone: \(V = \tfrac{1}{3}\pi r^2 h\)
• Sphere: \(V = \tfrac{4}{3}\pi r^3\)

Error analysis: a student finds a cone's volume as \(\pi r^2 h = 3.14 \times 9 \times 4 = 113.04\). They forgot the \(\tfrac{1}{3}\); the real volume is \(\tfrac{1}{3}(113.04) = 37.68 \text{ cm}^3\).

Case study - a silo shaped like a cylinder with a cone on top: find each volume separately, then add. Break a complex solid into simple ones.

Final habit: name the solid and say "fill or cover" before you compute. The right formula does most of the work.

Formula Selection as a Decision Process

The formula sheet is most useful after you identify the target and the solid. Do not scan formulas for one containing familiar numbers. Follow a decision path: interior or exterior, shape, required measurements, then formula.

Mixed Worked Example: A Silo

A silo consists of a cylinder of radius \(3\text{ m}\) and height \(8\text{ m}\), topped by a cone with the same radius and height \(2\text{ m}\). Find total capacity.

1. Split the composite solid into non-overlapping pieces.
2. Cylinder: \(V=\pi(3)^2(8)=72\pi\).
3. Cone: \(V=\tfrac13\pi(3)^2(2)=6\pi\).
4. Add: \(V_{\text{total}}=78\pi\approx245.0\text{ m}^3\).

The circular face where the pieces meet is internal. That matters for surface area, but not for volume: adding the two non-overlapping interior spaces is correct.

Final Error Audit

Before submitting, inspect five things: Did you use radius rather than diameter? Did you apply the exponent to the radius or side before multiplying? Did you include ( frac13) for a cone or ( frac43) for a sphere? Did you count only exposed faces for surface area? Did you report square or cubic units as required?

Mastery Challenge

Create a one-page memory map from meaning rather than copied formulas. Put fill/hold on one side and cover/wrap on the other. Under each, sketch a box, cylinder, cone, and sphere and write the corresponding formula structure. Beside every formula, record its most likely error. A useful review sheet should help you make decisions, not merely display symbols.

GED® Coaching: A Reliable Five-Pass Method

Use the same disciplined process on every three-dimensional measurement problem.

1. Name the target. Write volume or surface area before calculating. Words such as hold, fill, capacity, and inside usually signal volume. Words such as cover, wrap, paint, label, and outside usually signal surface area.
2. Identify the solid and the given measurements. Mark length, width, height, radius, diameter, base area, or slant height. Never assume a drawing is to scale. If a diameter is given but the formula uses radius, write \(r=d/2\) before substituting.
3. Write the formula in symbols. Use the GED® formula sheet, but choose the formula from the meaning of the problem. Then substitute values with units. This makes a missing square, cube, or one-third factor much easier to spot.
4. Calculate in visible stages. Evaluate exponents first, then multiplication and division. Keep \(\pi\) in the calculator until the final step unless the question directs you to use \(3.14\). Round only once, at the end.
5. Verify meaning, size, and unit. Volume must use cubic units; surface area must use square units. Estimate with friendly numbers and ask whether the answer fits the situation. A cone matching a cylinder must have one-third of its volume, and a closed container must include every exposed face.

When an answer is wrong, classify the cause rather than merely recording the score: target error, solid-identification error, formula error, radius/diameter error, exponent error, unit error, calculation error, or rounding error. Rewrite the solution from the first incorrect decision. This correction routine turns each missed item into a reusable GED® strategy.

Session Practice Routine

1. State whether the target is interior volume or exterior surface area.
2. Name the solid and mark only the required measurements.
3. Convert dimensions to compatible units before using a formula.
4. Write the formula, substitute visibly, and evaluate in stages.
5. Attach square or cubic units and verify the size by estimation.
6. Complete the interactive self-check, then explain any correction aloud.

Exit Reflection

Without looking back, sketch the session's solid or net, write the key formula
from meaning, name one likely error, and describe the check that would catch it.