📏 Area, Volume, Circles and Coordinates

Part 1 of 7 — Measurement on the ASVAB: Perimeter and Formulas

Topic Overview

What This Looks Like on the ASVAB

Mathematics Knowledge (MK) is the fifth subtest on the CAT-ASVAB: 15 scored questions in 31 minutes, so you have about 2 minutes per question. Every question has four answer choices, and there is no calculator. At MEPS you get two sheets of scratch paper and two pencils, and you can ask for more.

MK is one of the four subtests in your AFQT score (AR + MK + 2·VE), the score the military uses to check whether you are eligible to enlist. It also feeds several job line scores, such as the Air Force A and E composites.

Measurement questions are some of the most predictable points on MK. Expect questions that ask you to:

• find the perimeter or area of a shape that is drawn or described
• find the circumference or area of a circle, often with the answer written "in terms of $\pi$"
• find the volume of a box or cylinder (the official sample questions include the volume of a rectangular solid and the area of a square)
• work on the coordinate plane: slope, midpoint and distance

Key Fact: Plan on having no formulas given to you: learn the formulas on the next screen by heart. Once you know them, most of these questions take one or two steps.

📏 Area, Volume, Circles and Coordinates

Part 2 of 7 — Area of Polygons and Composite Figures

What Area Measures

Area is the number of 1-by-1 squares it takes to cover a flat shape. That is why the answer is always in square units: square feet, square meters, square inches.

The rule that decides most area questions

The height is always measured straight up from the base, at a right angle. In a drawing it is usually a dashed line with a small square at the bottom. A slanted side is not the height (unless the shape has a right angle there).

Why does a triangle get the $\frac{1}{2}$? Two copies of any triangle fit together into a parallelogram with the same base and height, so one triangle is half of $bh$.

📏 Area, Volume, Circles and Coordinates

Part 3 of 7 — Circles: Circumference and Area

The Parts of a Circle

• The radius $r$ runs from the center to the circle.
• The diameter $d$ runs all the way across through the center: $d = 2r$, and $r = \frac{d}{2}$.
• The circumference $C$ is the distance around the circle (its perimeter).
• $\pi$ (pi) is the number you get when you divide any circle's circumference by its diameter. It is a little more than 3: about $3.14$, or $\frac{22}{7}$.

"In terms of $\pi$"

Many ASVAB answer choices look like $16\pi$ or $64 - 16\pi$. That means leave $\pi$ as a symbol and do not multiply it out. It makes your work faster: a circle with radius 4 has area $\pi \cdot 4^2 = 16\pi$, done.

If the choices are plain numbers, use $3.14$ or $\frac{22}{7}$ (use $\frac{22}{7}$ when a number is a multiple of 7).

Key Fact: The area formula squares the radius. If the problem gives the diameter, cut it in half first.

📏 Area, Volume, Circles and Coordinates

Part 4 of 7 — Volume and Surface Area

Two Different Questions About a Solid

• Volume is how much space the solid fills: how many 1-by-1-by-1 cubes fit inside. Units are cubic: cubic feet (cu ft), cubic centimeters.
• Surface area is the total area of all the outside faces: how much paint or wrapping it takes. Units are square.

Where the formulas come from

Every one of these volumes is (area of the base) × (height). A box's base is a rectangle, $lw$; a cylinder's base is a circle, $\pi r^2$.

A box has 6 faces in 3 matching pairs: top and bottom ($lw$ each), front and back ($lh$ each), left and right ($wh$ each). A cube's 6 faces are all $s^2$.

A cylinder's surface is two circles ($\pi r^2$ each) plus the curved side. Unroll the side and it becomes a rectangle: one side is the height $h$, the other is the circumference $2\pi r$. So the side's area is $2\pi r h$.

📏 Area, Volume, Circles and Coordinates

Part 5 of 7 — The Coordinate Plane: Slope, Midpoint and Distance

Points and Quadrants

The coordinate plane is a grid made by two number lines: the horizontal x-axis and the vertical y-axis. They cross at the origin, $(0, 0)$.

Every point is an ordered pair $(x, y)$: first move left or right ($x$), then up or down ($y$). The point $(-4, 1)$ is 4 left and 1 up.

The axes split the plane into four quadrants, numbered counterclockwise starting at the upper right:

A point on an axis, such as $(0, 5)$ or $(-2, 0)$, is not in any quadrant.

📏 Area, Volume, Circles and Coordinates

Part 6 of 7 — Test Strategy and Common Traps

Your Clock on MK

On the CAT-ASVAB, MK gives you 31 minutes for 15 scored questions: about 2 minutes each. A measurement question you know cold takes well under a minute, which banks time for harder ones.

Three rules of the computer test shape your strategy:

• You can't go back. Once you answer, the question is gone. Check your work on scratch paper before you click.
• The test adapts. A right answer usually brings a harder question next. Harder measurement questions combine steps: a shaded region, a working-backward problem, a unit change.
• Don't rush the end. If time runs short, keep reading and genuinely solving. The CAT applies a relatively large penalty when several wrong answers pile up at the end of a subtest, so random clicking hurts. Some questions may be unscored tryouts, but you can't tell which, so treat every one as scored.

A four-step routine for every measurement question

1. Name what is asked: perimeter (length), area (square units), volume (cubic units), or a coordinate idea.
2. Write the formula from memory on scratch paper before plugging in.
3. Check each number's job: radius or diameter? height or slanted side? feet or yards?
4. Sanity-check: is the answer the right size and in the right units? Use $\pi \approx 3$ to estimate.

📏 Area, Volume, Circles and Coordinates

Part 7 of 7 — Mixed Mastery Set

How This Set Works

Twelve questions from the whole topic, in the same mix the computer test gives you: a few quick ones, a solid middle, and some multi-step questions at the end.

• Give yourself about 2 minutes per question, the CAT-ASVAB pace for MK.
• No calculator. Use scratch paper.
• Run the routine every time: name what is asked, write the formula, check each number's job, sanity-check the answer.

A missed question points you back to its part: perimeter and formulas (Part 1), area (Part 2), circles (Part 3), volume and surface area (Part 4), the coordinate plane (Part 5), traps (Part 6).