8.1 Area, Volume, and Composite Figures
Key Takeaways
- GED Math geometry questions test choosing the correct measurement (perimeter, area, volume, or surface area) before substituting numbers into a formula-sheet formula.
- Perimeter and circumference use linear units, area and surface area use square units, and volume uses cubic units, so the unit is your first error-check.
- Composite 2-D figures should be split into rectangles, triangles, circles, or trapezoids, then added or subtracted depending on whether the prompt says 'remaining', 'shaded', or 'total'.
- Composite 3-D figures use the same logic: compute each prism, cylinder, cone, or pyramid separately, then combine only the parts the question requests.
- The provided formula sheet supplies formulas but not judgment; candidates must still separate radius from diameter, perpendicular height from slant height, and area from surface area.
Start With What Is Being Measured
The GED Mathematical Reasoning test is a single 115-minute section of 46 items: 5 questions in a short no-calculator part followed by 41 questions where the on-screen TI-30XS MultiView calculator and a formula sheet are available. You pass the section with a scaled score of 145 on the 100-200 scale. Roughly 20-30% of items involve geometry and measurement, and they are almost always framed as practical tasks: tile a floor, fence a yard, fill a tank, paint a box, or compare package sizes.
The first decision is never which numbers to multiply. The first decision is what kind of measurement the question wants. Use the unit as a warning signal. Perimeter and circumference measure the distance around an object, so the answer is in feet, inches, or meters. Area measures flat coverage and uses square units. Volume measures inside capacity and uses cubic units. Surface area measures the outside covering of a solid, so it uses square units even though the object is three-dimensional.
GED Geometry Decision Table
| Prompt language | Likely measure | Unit check |
|---|---|---|
| Fence, trim, border, distance around | Perimeter or circumference | Linear units (ft, in, m) |
| Carpet, paint one wall, land, shaded region | Area | Square units |
| Water, storage, air, soil, concrete inside | Volume | Cubic units |
| Wrapping, outside paint on a box, label on a can | Surface area | Square units |
| Missing side in a right triangle | Pythagorean theorem | Linear units |
Reading the unit in the answer choices is itself a strategy: if three choices are in square feet and one is in cubic feet, the test is signaling which measure it expects.
Composite Figures
A composite figure is built from smaller familiar shapes. On GED Math, this may be an L-shaped room, a rectangle with a semicircle attached, a region with a rectangular cutout, or a solid made from two stacked prisms. The reliable method has four moves: draw the boundary, label the simple parts, compute each part, then decide whether the prompt wants a total (add) or a leftover (subtract).
Worked 2-D example. A playground is a 30 ft by 18 ft rectangle with a 10 ft by 6 ft sandbox removed. The full rectangle is 30 * 18 = 540 square feet. The sandbox is 10 * 6 = 60 square feet. Because the word is removed, subtract: 540 - 60 = 480 square feet of play surface.
Worked 3-D example. A storage container is a rectangular prism 8 ft long, 4 ft wide, 3 ft high, with a second prism 4 ft by 4 ft by 2 ft on top. The bottom volume is 8 * 4 * 3 = 96 cubic feet; the top is 4 * 4 * 2 = 32 cubic feet. The total interior capacity is 96 + 32 = 128 cubic feet.
Formulas You Will Reach For
- Rectangle area: A = lw; triangle area: A = (1/2)bh
- Trapezoid area: A = (1/2)(b1 + b2)h
- Circle: area = pi * r^2; circumference = 2 * pi * r (or pi * d)
- Rectangular prism volume: V = lwh; cylinder volume: V = pi * r^2 * h
- Cone volume: V = (1/3) * pi * r^2 * h; pyramid: V = (1/3) * (base area) * h; sphere: V = (4/3) * pi * r^3
Formula-Sheet Traps
The formula sheet does not choose the formula for you. A circle formula needs the radius, so if the prompt gives a diameter, halve it first. A cylinder volume needs base area times height, not circumference times height. A surface-area problem may count only exposed faces if one side sits against a wall or another solid. Notice the one-third in cone and pyramid volume: a common trap answer is exactly three times the right value because the test-taker forgot it.
Worked Mini-Example and the Reasonableness Check
A circular garden has diameter 12 ft and is surrounded by a rectangular walkway region that measures 18 ft by 16 ft overall. How much walkway area lies outside the garden? The rectangle area is 18 * 16 = 288 square feet. The garden radius is 12 / 2 = 6 ft, so the circle area is about 3.14 * 6^2 = 3.14 * 36 = 113.04 square feet. The walkway area is 288 - 113.04 = 174.96 square feet, which rounds to about 175 square feet.
Before selecting an answer, label the result and check it for reasonableness. If the prompt asks for area and your number is in feet rather than square feet, your process slipped somewhere. If a volume answer comes out smaller than the matching floor area even though the height is greater than 1, you probably dropped the height. If a circle area is larger than the square that would enclose it, you likely used the diameter as the radius.
These checks are fast and very GED-specific, because the wrong answer choices are deliberately built from common formula and unit mistakes such as forgetting to square the radius or doubling instead of halving the diameter. Spend the last ten seconds of each geometry item confirming the unit and rough size before you move on.
A rectangular kitchen floor is 16 feet long and 11 feet wide. A rectangular island that is 5 feet by 3 feet will not be tiled. How many square feet of tile are needed for the exposed floor?
A cylinder has radius 4 inches and height 9 inches. Which expression gives its volume?