10.1 Area, Perimeter, and Prism Volume
Key Takeaways
- AAF weights Geometry concepts for Algebra 1 at 5–10% of the 20-item CAT, typically 1–2 questions covering area, perimeter, and volume; distance and Pythagorean theorem; and geometric transformations.
- A 6 cm by 5 cm by 2 cm right rectangular prism has volume V = 60 cm³ and surface area SA = 2(30 + 12 + 10) = 104 cm²; 60 is the complementary trap when the stem asks for surface area.
- Rectangle area is ℓw and perimeter is 2(ℓ + w); triangle area is (1/2)bh with h perpendicular to that base; circle area is πr² and circumference is 2πr.
- If length is x + 2, width is x, and height is 4, volume is 4x(x + 2) and surface area is 2x² + 20x + 16.
- Table 11 puts area, perimeter, and prism volume with algebraic dimensions in Geometry concepts for Algebra 1; cylinder, cone, and sphere volume belong to Algebra 2 geometry.
10.1 Area, Perimeter, and Prism Volume
College Board’s Next-Generation ACCUPLACER Advanced Algebra and Functions (AAF) test weights Geometry concepts for Algebra 1 at 5–10% of the 20-item computer-adaptive test — typically 1–2 items. Table 11 of the Next-Generation ACCUPLACER Test Specifications Manual splits that slice into three skills: creating expressions for area, perimeter, and volume; using the distance formula and Pythagorean theorem; and evaluating dilations, rotations, translations, and reflections. This section is the first skill: write and evaluate formulas for plane figures and for a right rectangular prism.
The student-facing What’s on the Tests page currently names nine AAF categories and omits geometry. Table 10 still includes it, and College Board’s AAF Sample Questions PDF lists Geometry concepts among the knowledge/skill categories. Treat the 5–10% as live. A CAT that never serves geometry is luck, not a plan.
The formulas you must write, not just recall
AAF rarely asks you to “state the formula.” It asks you to create an expression in numbers or in x. Lock the formulas, then substitute.
| Figure | Perimeter / circumference | Area |
|---|---|---|
| Rectangle, sides ℓ and w | P = 2(ℓ + w) | A = ℓw |
| Square, side s | P = 4s | A = s² |
| Triangle, sides a, b, c; base b and height h | P = a + b + c | A = (1/2)bh |
| Circle, radius r | C = 2πr | A = πr² |
Perimeter is the total length of the boundary (one-dimensional: cm, ft, or x units). Area is the measure of the interior (two-dimensional: cm², ft²). Mixing those units is a scoring trap: an option 28 cm cannot be the area of a 7-by-4 rectangle.
A right rectangular prism (a box) has three pairwise perpendicular dimensions: length ℓ, width w, and height h. Two quantities appear on AAF:
- Volume:
V = ℓwh(cubic units). How much the box holds. - Surface area:
SA = 2(ℓw + ℓh + wh)(square units). The total area of all six faces.
The factor of 2 is there because each pair of faces appears twice: two ℓ-by-w bases, two ℓ-by-h sides, and two w-by-h sides. Dropping the 2 reports three faces, not six.
Worked example: rectangle and triangle
A rectangular patio is 7 ft by 4 ft.
- Area:
A = 7 · 4 = 28 ft². - Perimeter:
P = 2(7 + 4) = 22 ft.
If the same patio is “3 more than twice the width,” with width w, then length is 2w + 3. The area expression is (2w + 3)w = 2w² + 3w. The perimeter expression is 2((2w + 3) + w) = 2(3w + 3) = 6w + 6. AAF will ask for one of those expressions, not both. Read the stem: area versus perimeter is the first filter.
A triangle has base 10 cm and corresponding height 6 cm. Area is (1/2)(10)(6) = 30 cm². The word corresponding matters: height is the perpendicular distance to that base. If the three sides are 5 cm, 12 cm, and 13 cm, the perimeter is 5 + 12 + 13 = 30 cm. That 5-12-13 triangle is right-angled (Section 10.2), so you may also take the legs as base and height: (1/2)(5)(12) = 30 cm² again. Using a non-perpendicular side as “height” is the standard miss.
If an isosceles triangle has two equal sides of length 2x + 1 and base x, the perimeter is (2x + 1) + (2x + 1) + x = 5x + 2. Area still needs a height; do not treat a slanted side as h.
A square of side 2x + 1 has perimeter 4(2x + 1) = 8x + 4 and area (2x + 1)² = 4x² + 4x + 1. Expanding the square as 4x² + 1 (the dropped middle term) is the same algebra error as in the quadratics chapter.
Worked example: circle
A circle has radius 6 in. Circumference is 2π(6) = 12π in. Area is π(6)² = 36π in². Leave answers in terms of π unless the stem asks for a decimal.
The diameter is 12 in. Substituting 12 for r is the classic error: that would produce circumference 24π and area 144π, both wrong — the area is four times too large because radius is squared. If the stem gives diameter d, then r = d/2, C = πd, and A = π(d/2)² = πd²/4.
If a circular track has radius x + 3, the circumference expression is 2π(x + 3) = 2πx + 6π and the area expression is π(x + 3)² = π(x² + 6x + 9). Expand only when the options are expanded.
Worked example: original prism 6 cm × 5 cm × 2 cm
A closed cardboard box measures 6 cm by 5 cm by 2 cm. These are not College Board’s sample dimensions. Memorize the method, not a particular triple.
Volume:
V = 6 · 5 · 2 = 60 cm³.
Surface area, face by face:
- two 6-by-5 faces:
2 · 30 = 60 - two 6-by-2 faces:
2 · 12 = 24 - two 5-by-2 faces:
2 · 10 = 20
Total SA = 60 + 24 + 20 = 104 cm².
Compact formula:
SA = 2(ℓw + ℓh + wh) = 2(6 · 5 + 6 · 2 + 5 · 2) = 2(30 + 12 + 10) = 2(52) = 104 cm².
The trap: 60 looks like a tidy multiple-choice number, and it is sitting in your scratch work as the volume. If the stem asked for surface area, 60 cm³ is wrong on two counts — the quantity and the unit. If the stem asked for volume, 104 cm² is the complementary mistake. AAF options often include both numbers with swapped units.
An open-top box would drop the top ℓ-by-w face: SA_open = 2(ℓh + wh) + ℓw. Do not use the closed-box formula if the stem says “no lid.”
Check the arithmetic another way. The sum of the three unique face areas is 30 + 12 + 10 = 52. Doubling that 52 is the closed surface. If you stop at 52, you have painted three faces.
Algebraic dimensions: length = x + 2
Let a right rectangular prism have length x + 2, width x, and height 4.
Volume:
V = (x + 2)(x)(4) = 4x(x + 2) = 4x² + 8x.
Surface area:
SA = 2[(x + 2)x + (x + 2)(4) + (x)(4)]
= 2[x² + 2x + 4x + 8 + 4x]
= 2[x² + 10x + 8]
= 2x² + 20x + 16.
If the item asks for volume in factored form, 4x(x + 2) is finished. If the options are expanded, 4x² + 8x is finished. Matching the form of the options is part of the skill.
A rectangular garden that is 3x by x + 5 has perimeter 2(3x + x + 5) = 2(4x + 5) = 8x + 10 and area 3x(x + 5) = 3x² + 15x. Expanding 3x(x + 5) as 3x² + 5 (the dropped distribution) is the same algebra error as in linear equations.
A composite L-shaped floor plan is two rectangles, not a new formula. A 10-by-8 rectangle with a 4-by-3 rectangle removed has area 80 − 12 = 68. Perimeter of an L is not the sum of the two perimeters — shared interior edges are not on the boundary. Walk the outer path edge by edge.
AAF traps for this slice
- Reporting volume when asked for surface area (or the reverse). Compute both, then match the stem and the units.
- Forgetting the 2 in
SA = 2(ℓw + ℓh + wh), which reports three faces. - Using diameter as radius in
πr². - Triangle area without a perpendicular height. Side length is not automatically
h. - Perimeter of a rectangle as
ℓ + winstead of2(ℓ + w). - Adding area and perimeter into one expression. They have different dimensions and never combine.
- Cylinder, cone, or sphere formulas on an Algebra 1 geometry item. Those belong to nonprism-volume.
College Board’s calculator policy: some AAF items show an on-screen calculator icon; handheld calculators are not allowed except with an approved accommodation. Area and surface-area arithmetic is faster by hand if the dimensions are small integers. Algebraic dimensions never need a calculator.
A closed right rectangular prism measures 6 cm by 5 cm by 2 cm. What is its surface area?
A rectangle has length x + 2 and width x. Which expression is its perimeter?
A right rectangular prism has length x + 2, width x, and height 3. Which expression is its volume?
A closed right rectangular prism is 8 cm long, 3 cm wide, and 5 cm high. Which value is its volume?