Area, Volume, Circles and Coordinates

📏 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.

The Measurement Formulas to Memorize

Perimeter and area

Volume and surface area

The coordinate plane

Units

Perimeter is a length (ft), area is in square units (sq ft), and volume is in cubic units (cu ft).

Warm-up. 🎯 Three quick ones to see where you stand. No calculator.

Perimeter: The Distance Around

Perimeter is the total length of the boundary. For any polygon, add the lengths of all the sides. The formulas are shortcuts for that:

• Rectangle: two lengths and two widths, $P = 2l + 2w$
• Square: four equal sides, $P = 4s$
• Regular polygon (all sides equal): number of sides × side length

Worked examples

1. Fencing a yard. A rectangular storage yard is 45 ft by 30 ft. How much fence goes around it?

$P = 2(45) + 2(30) = 90 + 60 = 150$ ft.

2. Working backward from the perimeter. A rectangle has a perimeter of 50 m and a length of 15 m. Find the width.

The two lengths use $2 \times 15 = 30$ m, which leaves $50 - 30 = 20$ m for the two widths, so $w = 10$ m.

3. A regular hexagon. Each of the 6 sides is 8 in, so $P = 6 \times 8 = 48$ in.

4. Square from its perimeter. A square has perimeter 36 ft, so each side is $36 \div 4 = 9$ ft (and its area is $9^2 = 81$ sq ft).

Shapes with all right angles

When every corner of a shape is a right angle (an L-shape or a step shape), you can find missing sides with one idea: the short pieces on one side add up to the long side across from them.

Example: an L-shape has a bottom of 12 ft and a left side of 7 ft. The horizontal pieces across the top must add up to 12 ft, and the vertical pieces on the right must add up to 7 ft. So the perimeter is the same as a 12 by 7 rectangle:

$P = 2(12) + 2(7) = 38$ ft.

Key Fact: Perimeter never uses multiplication of two sides. If you multiplied a length by a width, you found an area.

Find the missing sides. 🔽 Use the figure. Every corner is a right angle.

Perimeter of the pad. 🎯 Use the figure.

Perimeter in context. 🎯

Perimeter practice. ✍️ Enter numbers only.

1) The perimeter, in centimeters, of a rectangle 14 cm long and 6 cm wide:

2) A square has a perimeter of 52 m. The length of one side, in meters:

3) An equilateral triangle (three equal sides) has a perimeter of 27 ft. The length of one side, in feet:

Key Takeaways — Part 1

• MK on the CAT-ASVAB: 15 questions, 31 minutes, about 2 minutes each, four choices, no calculator
• Memorize the formula tables: no formulas are handed to you
• Perimeter = add every side; rectangle $2l + 2w$, square $4s$, regular polygon $n \times s$
• Work backward: subtract what you know from the perimeter, then divide
• In an all-right-angle shape, the short pieces on one side add up to the long side across from them
• Perimeter is in plain units (ft), area in square units, volume in cubic units

Up Next: Part 2 — area of rectangles, triangles, parallelograms, trapezoids and shapes built from them.

📏 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 $bh$.

Worked Examples

1. Rectangle. A supply room floor is 12 ft by 7 ft.

$A = 12 \times 7 = 84$ sq ft.

2. Triangle. Base 9 in, height 6 in.

$A = \frac{1}{2}(9)(6) = \frac{1}{2}(54) = 27$ sq in. Tip: halve whichever number is even first: $9 \times 3 = 27$.

3. Parallelogram. Base 11 m, height 5 m, slanted side 6 m.

$A = bh = 11 \times 5 = 55$ sq m. The 6 m side is not used.

4. Trapezoid. Parallel bases 5 ft and 9 ft, height 4 ft.

Add the bases: $5 + 9 = 14$. Halve: $7$. Multiply by the height: $7 \times 4 = 28$ sq ft.

5. Working backward. A triangle has an area of 30 sq cm and a base of 12 cm. Find the height.

$30 = \frac{1}{2}(12)h = 6h$, so $h = 5$ cm. Remember to undo the $\frac{1}{2}$: the height is not $30 \div 12$.

6. Square from its area. A square patio covers 64 sq ft. Each side is $\sqrt{64} = 8$ ft.

Triangle area. 🎯 Use Figure 1.

Parallelogram area. 🎯 Use Figure 2.

Trapezoid area. 🎯 Use Figure 3.

Composite Figures: Add or Subtract

A composite figure is built from shapes you already know. There are two ways in:

• Add: cut the figure into rectangles and triangles, find each area, and add.
• Subtract: find the area of a big shape and take away the piece that is missing.

Worked examples

An L-shaped room. A room is a 10 ft by 6 ft rectangle with a 4 ft by 2 ft closet corner cut out of it.

Subtract: $10 \times 6 - 4 \times 2 = 60 - 8 = 52$ sq ft.

A picture frame. An 8 in by 10 in photo sits in a frame whose outside edge is 12 in by 14 in. How much area is the frame itself?

Big minus small: $12 \times 14 - 8 \times 10 = 168 - 80 = 88$ sq in.

A house-shaped sign. A 6 ft by 4 ft rectangle with a triangle on top: the triangle's base is 6 ft and its height is 3 ft.

Add: $6 \times 4 + \frac{1}{2}(6)(3) = 24 + 9 = 33$ sq ft.

Key Fact: Cutting a corner out of a rectangle lowers the area but leaves the perimeter unchanged: the two inner sides of the cut replace the two outer pieces that were removed.

A lot with a corner removed. ✍️ Use Figure 4. Enter numbers only.

1) The area of the shaded lot, in square meters:

2) The perimeter of the shaded lot, in meters:

Quick area calculations. 🔽

Area in context. 🎯

Key Takeaways — Part 2

• Area is in square units: rectangle $lw$, square $s^2$, parallelogram $bh$, triangle $\frac{1}{2}bh$, trapezoid $\frac{1}{2}(b_1 + b_2)h$
• The height meets the base at a right angle; a slanted side is a trap
• To work backward from a triangle's area, double the area before dividing by the base
• Composite figures: add the pieces or subtract the missing piece
• Cutting a rectangular corner out of a rectangle changes the area, not the perimeter

Up Next: Part 3 — circles: radius, diameter, circumference and area, with answers in terms of $\pi$.

📏 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.

Worked Examples

1. Given the radius. $r = 6$: $C = 2\pi(6) = 12\pi$ and $A = \pi(6^2) = 36\pi$.

2. Given the diameter. $d = 10$, so $r = 5$: $C = \pi(10) = 10\pi$ and $A = \pi(5^2) = 25\pi$.

3. Given the area. $A = 49\pi$. Then $r^2 = 49$, so $r = 7$, $d = 14$ and $C = 14\pi$.

4. Given the circumference. $C = 18\pi$. Then $d = 18$, so $r = 9$ and $A = 81\pi$.

5. Using a number for $\pi$. A wheel has a diameter of 2 ft. In one full turn it rolls one circumference: $C = \pi d \approx 3.14 \times 2 = 6.28$ ft.

Parts of circles

• A semicircle (half circle) has half the area: $\frac{1}{2}\pi r^2$.
• A quarter circle has a quarter of the area: $\frac{1}{4}\pi r^2$.
• The perimeter of a semicircular region is half the circumference plus the straight diameter. For $r = 5$: $\frac{1}{2}(10\pi) + 10 = 5\pi + 10$.

Shaded regions

"Find the shaded area" usually means big shape minus small shape. A ring between circles of radius 5 and radius 2 has area $25\pi - 4\pi = 21\pi$.

Read the radius. ✍️ Use Figure 1. Enter numbers only.

1) The diameter of the circle, in centimeters:

2) The circumference is ___$\pi$ cm. Enter the number in front of $\pi$:

3) The area is ___$\pi$ square cm. Enter the number in front of $\pi$:

Diameter given. 🎯 Use Figure 2.

Shaded corners. 🎯 Use Figure 3.

A window. 🎯 Use Figure 4.

How circles grow. 🔽

Circles in context. 🎯

Key Takeaways — Part 3

• $d = 2r$; circumference $C = 2\pi r = \pi d$; area $A = \pi r^2$
• If you are given the diameter, halve it before squaring
• "In terms of $\pi$": keep $\pi$ as a symbol ($25\pi$, $64 - 16\pi$)
• Use $3.14$ or $\frac{22}{7}$ only when the choices are plain numbers
• Semicircle area $\frac{1}{2}\pi r^2$; a semicircular region's perimeter adds the diameter
• Shaded region = big shape minus small shape
• Double the radius: circumference doubles, area quadruples

Up Next: Part 4 — volume and surface area of boxes, cubes and cylinders.

📏 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 $2\pi r h$.

Worked Examples

1. A footlocker. 3 ft long, 2 ft wide, 1.5 ft tall.

$V = 3 \times 2 \times 1.5 = 6 \times 1.5 = 9$ cu ft.

2. A cube with edge 4 cm.

$V = 4^3 = 64$ cubic cm and $SA = 6 \times 4^2 = 6 \times 16 = 96$ square cm.

3. A cylinder with radius 2 m and height 5 m.

$V = \pi(2^2)(5) = 20\pi$ cubic m. $SA = 2\pi(2^2) + 2\pi(2)(5) = 8\pi + 20\pi = 28\pi$ square m.

4. Working backward. A box holds 96 cubic in. It is 8 in long and 4 in wide. How tall is it?

$96 = 8 \times 4 \times h = 32h$, so $h = 3$ in.

5. Cubic yards. A concrete slab is 9 ft by 6 ft by 0.5 ft: $V = 27$ cu ft. Since 1 cu yd $= 3 \times 3 \times 3 = 27$ cu ft, that is exactly 1 cubic yard.

6. Scaling. If every edge of a cube is doubled, each face's area is multiplied by $2^2 = 4$ and the volume by $2^3 = 8$.

Key Fact: Read the question for the word that matters: holds, fills, capacity → volume. Paint, cover, wrap → surface area.

Volume of a box. 🎯 Use Figure 1.

Surface area of the box. ✍️ Use Figure 1. Enter numbers only.

1) The area of the top face (8 in by 5 in), in square inches:

2) The area of the front face (8 in by 3 in), in square inches:

3) The total surface area of the closed box, in square inches:

Volume of a cylinder. 🎯 Use Figure 2.

Surface area of the cylinder. ✍️ Use Figure 2. Enter the number in front of $\pi$.

1) The area of the curved side is ___$\pi$ square cm:

2) The total surface area of the closed cylinder is ___$\pi$ square cm:

Cubes. 🔽

Volume in context. 🎯

Key Takeaways — Part 4

• Volume = base area × height: box $lwh$, cube $s^3$, cylinder $\pi r^2 h$ (cubic units)
• Surface area = sum of all faces: box $2lw + 2lh + 2wh$, cube $6s^2$, cylinder $2\pi r^2 + 2\pi r h$ (square units)
• The curved side of a cylinder unrolls into a rectangle: circumference × height
• 1 cu yd = 27 cu ft (not 9, not 3)
• Scale every length by $k$: areas grow by $k^2$, volumes by $k^3$
• Holds or fills → volume; paint or wrap → surface area

Up Next: Part 5 — the coordinate plane: points, slope, midpoint and distance.

📏 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.

Reading points. 🎯 Use Figure 1.

Slope: How Steep a Line Is

Slope $m$ is rise over run: how much $y$ changes for each step in $x$.

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$

Example 1. Through $(1, 2)$ and $(5, 10)$: $m = \frac{10 - 2}{5 - 1} = \frac{8}{4} = 2$.

Example 2. Through $(-2, 7)$ and $(3, -3)$: $m = \frac{-3 - 7}{3 - (-2)} = \frac{-10}{5} = -2$. Subtract in the same order on top and bottom.

Slope-intercept form: $y = mx + b$

In $y = mx + b$, the number in front of $x$ is the slope and $b$ is the y-intercept, where the line crosses the y-axis at $(0, b)$.

Example 3. $y = -2x + 3$ has slope $-2$ and crosses the y-axis at $(0, 3)$.

Example 4. Rewrite $6x + 2y = 8$: subtract $6x$ to get $2y = -6x + 8$, then divide by 2: $y = -3x + 4$. Slope $-3$, y-intercept $4$.

Parallel lines have the same slope. (Perpendicular lines have slopes that are negative reciprocals, such as $2$ and $-\frac{1}{2}$.)

Slope from a graph. 🎯 Use Figure 2.

Slope practice. 🔽

Midpoint and Distance

Midpoint: average the coordinates

$\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Example. The midpoint of $(-2, 4)$ and $(6, 0)$ is $\left(\frac{-2 + 6}{2}, \frac{4 + 0}{2}\right) = (2, 2)$.

Distance: the Pythagorean theorem on a grid

Draw a right triangle under the segment. The legs are the change in $x$ and the change in $y$; the segment is the hypotenuse:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Example. From $(-1, 1)$ to $(5, 9)$: the legs are $5 - (-1) = 6$ and $9 - 1 = 8$, so $d = \sqrt{36 + 64} = \sqrt{100} = 10$.

Look for the familiar right-triangle side sets 3-4-5, 6-8-10, 5-12-13 and 8-15-17: they let you skip the square root.

Distance from P to Q. ✍️ Use Figure 3. Enter numbers only.

1) The length of the horizontal dashed leg:

2) The length of the vertical dashed leg:

3) The distance from $P$ to $Q$:

Midpoint and parallel lines. 🎯

Key Takeaways — Part 5

• $(x, y)$: across first, then up or down; quadrants I to IV go counterclockwise from the upper right
• Slope $= \frac{y_2 - y_1}{x_2 - x_1}$, rise over run, subtracting in the same order
• Horizontal line: slope $0$; vertical line: slope undefined
• $y = mx + b$: $m$ is the slope, $b$ is the y-intercept; solve for $y$ first when the equation is in another form
• Parallel lines share a slope
• Midpoint = average of the coordinates; distance = Pythagorean theorem on the grid

Up Next: Part 6 — test strategy and the traps that catch people on measurement questions.

📏 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.

The Traps the Wrong Answers Are Built From

The three wrong choices are rarely random. Each one is usually the result of a specific mistake. Know the mistakes and you can spot the traps.

Estimate to eliminate

When the choices are plain numbers, a quick estimate often leaves only one choice standing. A circle with radius 10 has area $\pi \times 100$, a bit more than $300$. If the choices are $31.4$, $62.8$, $314$ and $628$, only $314$ is close.

When the figure is only described

If the problem describes a shape in words, sketch it and label every number. Thirty seconds of drawing prevents most of the traps above.

Spot the mistake. 🎯 Each question shows a wrong answer and asks where it came from, or asks you to dodge the trap.

Name the measurement. 🔽 Step 1 of the routine: what is the question really asking for?

Estimate first. 🎯 Round the numbers (and use $\pi \approx 3$ when a circle is involved) to rule out choices, then confirm.

A patio job, start to finish. ✍️ A rectangular patio is 18 ft long and 12 ft wide. Concrete will be poured 6 inches (0.5 ft) thick. Enter numbers only.

1) The area of the patio, in square feet:

2) The area of the patio, in square yards:

3) The volume of concrete, in cubic yards:

Key Takeaways — Part 6

• About 2 minutes per MK question; you can't go back, so check before you click
• Routine: name what is asked, write the formula, check each number's job, sanity-check size and units
• Wrong choices come from predictable mistakes: diameter for radius, slanted side for height, missing $\frac{1}{2}$, perimeter vs area, $\div 3$ instead of $\div 9$ or $\div 27$, scaling not squared or cubed, slope order
• Estimate with $\pi \approx 3$ and round numbers to eliminate choices
• Sketch any figure that is only described in words

Up Next: Part 7 — a mixed mastery set at ASVAB difficulty.

📏 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).

Set A: quick ones. 🎯

Set B: the solid middle. 🎯

Set C: a composite figure. 🎯 Use Figure 1.

Set D: multi-step. 🎯

Key Takeaways — Part 7 and the Whole Topic

• Perimeter adds sides; area is in square units; volume is in cubic units
• Area: $lw$, $s^2$, $bh$, $\frac{1}{2}bh$, $\frac{1}{2}(b_1 + b_2)h$, $\pi r^2$ — height at a right angle to the base
• Circles: $C = 2\pi r = \pi d$; halve the diameter before squaring; keep answers in terms of $\pi$
• Volume: $lwh$, $s^3$, $\pi r^2 h$; surface area adds every face
• Coordinates: slope $\frac{y_2 - y_1}{x_2 - x_1}$, $y = mx + b$, midpoint = average, distance = Pythagorean theorem
• Composite and shaded regions: add the pieces or subtract the missing piece
• Watch the traps: radius vs diameter, slanted side vs height, $\div 9$ for square yards and $\div 27$ for cubic yards

Next step: take the exit quiz for this topic. Any part you miss there is the part to review before test day.