Fitting Shapes Together

🧱 Fitting Shapes Together

Part 1 of 7: What Shape Items Look Like

The second kind of Assembling Objects question gives you a handful of loose pieces and asks which figure shows them fitted together into one shape. If you have done the Connecting Parts lesson, the habits carry over: same pieces, turned but not flipped, checked one at a time.

AO at a glance

Both kinds of question appear on the AO subtest, so be ready for either one.

This Lesson

Anatomy of a Shape Item

The top box holds several separate pieces (two to five in this course), each drawn at some angle.

The four answer figures each show a finished shape, with lines marking where the pieces meet.

The correct figure:

• uses exactly the pieces in the box: the same number, each the same shape and size;
• may show any piece turned to any angle;
• fits the pieces with edges that meet exactly: no gaps and no overlaps.

Wrong figures usually:

• have one piece too many or too few;
• swap in a piece of a different shape or size (a bigger triangle, a narrower wedge);
• cut the finished shape along different lines;
• show a lopsided piece flipped into its mirror image. As in the connector lesson, this course treats a flipped piece as a different piece, because official materials do not spell out a flipping rule and a turned piece is the safe reading.

Key Fact: You are not judging whether the finished shape looks nice. You are checking whether it is built from these pieces.

Worked Example: Your First Shape Item

The box holds two pieces: a square and an isosceles triangle exactly as wide as the square. The correct figure uses those two pieces, at the same sizes, with nothing added and nothing left out.

Check each figure:

1. Same number of pieces?
2. Is each piece the same shape and size as one in the box?
3. Do the edges meet exactly, with no gap and no overlap?

Answer: Figure 4. The box holds 2 pieces: a square and an isosceles triangle as wide as the square. Figure 4 uses exactly those pieces, only turned, and their edges meet with no gaps or overlaps. Figure 1 has a much taller, pointier triangle than the one in the box. Figure 2 uses a short rectangle instead of a square for the bottom piece. Figure 3 leaves out the isosceles triangle as wide as the square, so there is a gap and only 1 pieces.

Warm-up 1. Count the pieces, then compare sizes.

Warm-up 2. Name each piece in the box (half? quarter? third?) before you look.

Warm-up 3. The rules in one question.

Key Takeaways: Part 1

• Shape items: loose pieces in the top box, four finished figures to choose from
• The correct figure uses the same pieces (number, shape, size), turned as needed, with no gaps or overlaps
• Wrong figures add or drop a piece, change a piece's size or shape, cut differently, or flip a lopsided piece
• AO: last CAT subtest, 15 questions in 18 minutes; not in the AFQT; used by the Navy

Up Next: Part 2: counting pieces and matching each one by shape and size.

🧱 Fitting Shapes Together

Part 2 of 7: Counting and Matching Pieces

The fastest check on any shape item is a count. It takes two seconds and often eliminates one or two figures outright.

Step 1: Count

Count the pieces in the top box. Then count the pieces in each figure (count the regions between the lines). Any figure with a different number is wrong.

Step 2: Name each piece

Give every piece in the box a short name with a size cue:

• "small square," "long thin rectangle," "quarter circle," "big triangle with a square corner"
• If two pieces are identical, note that too: "two matching trapezoids."

Step 3: Match piece by piece

For each surviving figure, find a region for every piece on your list, one region per piece:

• Shape: same number of sides, same kinds of corners (square, sharp, curved).
• Size: compared with the other pieces. If the triangle in the box is about half as long as the rectangle, it must be about half as long in the figure too.

Area check (for harder items)

The pieces must exactly fill the finished shape, so their areas add up to its area. If the box has two half circles but a figure shows a full circle plus an extra wedge, something is wrong. You rarely need numbers; just ask, "Is there too much or too little shape here for these pieces?"

Key Fact: Count, name, match. A figure that fails the count never needs a closer look.

Worked Example: Count, Then Match

The box has five pieces: four long, thin rectangles that are all alike, and one small square. Any figure without exactly five pieces is out at once. Then check that the survivors use four matching rectangles and one small square.

Answer: Figure 1. The box holds 5 pieces: a 3-by-1 rectangle, a second 3-by-1 rectangle, a third 3-by-1 rectangle, a fourth 3-by-1 rectangle, and a small 2-by-2 square. Figure 1 uses exactly those pieces, only turned, and their edges meet with no gaps or overlaps. Figure 2 is four long bars with no small square in the middle. Figure 3 is four equal squares, not four rectangles around a square. Figure 4 leaves out the small 2-by-2 square, so there is a gap and only 4 pieces.

Check 1. How many pieces, and what shape is each one?

Check 2. Count first.

Check 3. Count, then compare the size of each piece with the box.

Check 4. Two pieces: a quarter circle and a corner piece with one curved side.

Which figure shows the pieces in the top box fitted together correctly? Type the figure number (1, 2, 3 or 4).

Check 5. Areas must add up. ✍️

A rectangle that measures 6 units by 4 units is cut into three pieces. Two of the pieces have areas of 9 square units and 7 square units.

1) The area of the whole rectangle, in square units:

2) The area of the third piece, in square units:

Key Takeaways: Part 2

• Count first: a different number of pieces = wrong figure
• Name each box piece with a size cue, and note any matching pairs
• Match one region per piece by shape and by size compared with the other pieces
• Areas add up: the pieces exactly fill the finished shape

Up Next: Part 3: pieces that are turned, and pieces that are flipped.

🧱 Fitting Shapes Together

Part 3 of 7: Turned Pieces: Rotation Without Flipping

In the finished figure, pieces are almost always turned to a new angle. That is allowed. A piece flipped over into its mirror image is not, in this course.

Which pieces can be flipped "invisibly"?

A piece with a mirror line (a line that splits it into two matching halves) looks the same flipped as it does turned, so a flip can never make it wrong:

• squares, rectangles, circles, half circles, quarter circles, wedges of a circle;
• isosceles and equilateral triangles;
• rhombuses (diamonds) and isosceles trapezoids.

A piece with no mirror line is lopsided, and its mirror image is a genuinely different piece:

• right triangles with three different side lengths (legs 4 and 2, for example);
• L shapes made of four or more squares, P shapes and other lopsided step-shaped pieces (a three-square L is the exception: it has a mirror line along its diagonal);
• right trapezoids (one square side and one slanted side);
• parallelograms that lean.

How to spot a flipped piece

Use one feature as a pointer:

• L shape: picture it with the long bar standing upright and the foot at the bottom. Does the foot stick out to the right or to the left? A turn can never change the answer; a flip always does.
• Right triangle: find the square corner. Walking clockwise from it, do you go along the long leg first or the short leg first?
• Right trapezoid: walking clockwise from the square side, do you meet the long edge or the short edge first?

Answer the question for the piece in the box, then for the same piece in the figure. Different answers mean a flip.

Picture each piece with its long bar upright and its foot at the bottom. In the box and in both turned copies, the foot sticks out to the right. In the flipped copy it sticks out to the left, and no turn can fix that.

Worked Example: Two L Shapes

Both pieces are L shapes made of four squares. Two answer figures will look right at a glance, so compare the pieces as they sit in each figure with the pieces in the box: an L can be turned to any angle, but if its foot points the opposite way after every possible turn, it has been flipped.

Answer: Figure 1. The box holds 2 pieces: an L-shaped piece made of four squares and a second L-shaped piece made of four squares. Figure 1 uses exactly those pieces, only turned, and their edges meet with no gaps or overlaps. Figure 2 is a mirror image: its L-shaped piece made of four squares is flipped over, and turning a piece can never undo a flip. Figure 3 is made of two straight bars, not two L shapes. Figure 4 pairs a long five-square L with a straight bar, so the pieces are different sizes.

Check 1. The triangle and the four-sided piece are both lopsided, so watch for a flip.

Check 2. Count the squares in each piece, then check which way the L's foot points.

Check 3. Each trapezoid has one square side and one slanted side. Where do they sit?

Check 4. Letters make good practice pieces.

Key Takeaways: Part 3

• Turning a piece is allowed; flipping it into a mirror image is not (in this course)
• Only lopsided pieces can be caught by a flip: scalene right triangles, L and P shapes, right trapezoids, leaning parallelograms
• Use one feature as a pointer (the foot of the L, the square corner of a triangle) and compare box vs figure

Up Next: Part 4: edges and corners, where pieces actually meet.

🧱 Fitting Shapes Together

Part 4 of 7: Matching Edges and Corners

When the count and the piece names do not settle an item, look at where pieces meet. Real pieces can only sit flush when their edges and corners agree.

Edges

• Two pieces that share an edge must have edges of the same length there. A triangle side that is shorter than the rectangle side it sits on leaves bare edge; a longer one hangs over.
• A slanted edge can only meet another slanted edge that leans at the same angle.
• A curved edge can only meet a curved edge of the same circle (more in Part 5).

Corners

Corners that meet at one point must fill the space around it exactly:

Two square corners side by side make a straight edge (90° + 90° = 180°). Four square corners around one point fill it completely (4 × 90° = 360°).

Using it on the test

You will rarely compute angles. Instead, ask quick yes/no questions:

• Does this piece's long edge run the full length of the edge it sits against?
• Does a sharp corner sit in a spot that needs a square corner?
• Is there a sliver of empty space, or a spot where two pieces overlap?

Only the left panel shows a flush fit. A triangle whose side is too short or too long for the rectangle cannot be the piece that fits there, even if everything else about the figure looks tidy.

Worked Example: Edges and Corners

The box has two small right triangles (legs 1 and 2) and a parallelogram. Each triangle's slanted side must lie exactly along a slanted side of the parallelogram, and their square corners must become the corners of the rectangle.

Answer: Figure 3. The box holds 3 pieces: a right triangle with legs 1 and 2, a parallelogram, and a second right triangle with legs 1 and 2. Figure 3 uses exactly those pieces, only turned, and their edges meet with no gaps or overlaps. Figure 1 has bigger triangles (legs 2 and 2), so its middle piece is a different parallelogram. Figure 2 is cut into two trapezoids, with no small triangles. Figure 4 is a mirror image: its right triangle with legs 1 and 2 is flipped over, and turning a piece can never undo a flip.

Check 1. Which edge of the small triangle touches the trapezoid, and is it the right length?

Check 2. All three box pieces are the same triangle with three equal sides.

Check 3. The two triangles are identical, and each has three different side lengths.

Check 4. Corners must fill the space. ✍️

1) Three pieces meet at one point inside a finished figure. Two of their corners measure 90° and 150°. How many degrees is the third corner?

2) Two pieces meet at a point on a straight edge of a finished figure. One corner measures 135°. How many degrees is the other?

Key Takeaways: Part 4

• Touching edges must be the same length, and slanted edges must lean the same way
• Corners around an inside point add to 360°; along a straight edge, 180°
• Square corners belong in square spots: the corners of a rectangle need 90° pieces
• Look for slivers of empty space and overlaps; either one means the figure is wrong

Up Next: Part 5: curved pieces, circle wedges and builds with four or five pieces.

🧱 Fitting Shapes Together

Part 5 of 7: Curves, Circles and Harder Builds

Circle wedges

A wedge cut from a circle is named by the angle at its point. A full circle is 360°, so:

When a circle is built from wedges, their angles must add up to 360°: a half plus two quarters works (180 + 90 + 90), and so do wedges of 90°, 120° and 150°.

Size check: all wedges from one circle have the same radius (the same straight-edge length). A wedge with longer straight edges came from a bigger circle and cannot belong.

Curved edges meeting

A curved edge can sit against another curved edge only if they are parts of the same-size circle, one bulging out and one curving in. A quarter circle fills the scooped-out corner of a square only when its radius equals the square's side.

Builds with four or five pieces

More pieces means more chances for a figure to slip in one wrong piece. Keep it systematic:

1. Count.
2. Place the biggest or most unusual piece first (it has the fewest places it can go).
3. Check the pieces next to it, edge by edge.
4. Do any flip checks last.

All four wedges come from circles of the same size, so their straight edges are the same length. Only the angle at the point changes.

Worked Example: Three Different Wedges

The box has three wedges cut from one circle: a quarter (90°, a square corner), a wider 120° wedge and a still wider 150° wedge. Together they make 90 + 120 + 150 = 360 degrees, a full circle. Any figure whose wedges are the wrong sizes is out.

Answer: Figure 4. The box holds 3 pieces: a quarter-circle wedge (90 degrees), a 120-degree wedge, and a 150-degree wedge. Figure 4 uses exactly those pieces, only turned, and their edges meet with no gaps or overlaps. Figure 1 has three equal wedges, but the wedges in the box are three different sizes. Figure 2 has two quarters and a half, so its biggest wedge is too big. Figure 3 has a narrow 60-degree wedge where the box has a quarter wedge.

Check 1. Two wedges make a half circle (180°). How big is each one?

Check 2. Five pieces this time. Match the four long pieces and the middle square.

Check 3. A rectangle and two wide triangles.

Check 4. Wedge arithmetic. ✍️

1) A circle is cut into three wedges. Two of them measure 100° and 140°. How many degrees is the third?

2) How many 45° wedges fit together to make a half circle?

Key Takeaways: Part 5

• Wedges are named by their angle: half 180°, third 120°, quarter 90°, sixth 60°
• Wedges that make a whole circle add to 360° and share the same radius
• Curved edges meet only when they come from the same-size circle
• Big builds: count, place the biggest or oddest piece first, then work outward

Up Next: Part 6: a routine and a time budget for shape items.

🧱 Fitting Shapes Together

Part 6 of 7: Strategy, Timing and Traps

Your time budget

The shape-item routine

1. Count (about 5 seconds): cross out every figure with the wrong number of pieces.
2. Biggest or oddest piece (about 15 seconds): find it in each survivor. Is it the right shape and size?
3. Edges and corners (about 20 seconds): do the other pieces sit flush, with no slivers or overlaps?
4. Flip check (only if needed): for lopsided pieces, compare one pointer feature (the foot of the L, the square corner of a triangle).

If two minutes go by, pick the best survivor and move on.

CAT rules to remember

• No going back: finish your checks before you answer.
• Answer every item to move on; there is no skip-and-return on the CAT.
• Short on time? Keep making real choices. Official guidance says the CAT penalizes several wrong answers late in a subtest relatively heavily, and unfinished questions are scored as random guesses. On the paper test there is no guessing penalty, so fill in every answer.
• Scratch paper is provided at MEPS (two sheets and two pencils on the desk; click HELP and raise your hand for more). For a stubborn item, a ten-second sketch of the biggest piece can help, but most items need none.

Traps to watch for

Timed practice 1. Aim for about 70 seconds: count, biggest piece, edges, flips.

Timed practice 2. Same routine.

Check 3. Routine and rules. 🔽

Key Takeaways: Part 6

• CAT pace about 72 seconds per question; paper about 36
• Routine: count → biggest piece → edges and corners → flip check
• Match pieces, not outlines; a wrong figure can look perfectly finished
• No going back on the CAT; when time is short, keep making real choices

Up Next: Part 7: a ten-question mastery check.

🧱 Fitting Shapes Together

Part 7 of 7: Mastery Check

Ten shape items in a mix of difficulty. Give yourself about 12 minutes and use the routine every time:

1. Count the pieces.
2. Biggest or oddest piece: right shape, right size?
3. Edges and corners: flush, with no slivers or overlaps?
4. Flip check for lopsided pieces.

Read every explanation, including the ones you got right.

Mastery 1 of 10.

Mastery 2 of 10.

Mastery 3 of 10.

Mastery 4 of 10.

Mastery 5 of 10.

Mastery 6 of 10.

Mastery 7 of 10.

Mastery 8 of 10.

Mastery 9 of 10.

Mastery 10 of 10.

Fitting Shapes Together: Summary 🎓

The Shape-Item Checklist

1. Count: same number of pieces as the box
2. Name and match: every piece the same shape and size, one region each
3. Edges: same lengths where pieces touch; curves only against the same-size curve
4. Corners: 360° around an inside point, 180° along a straight edge
5. Flips: a lopsided piece mirrored is a different piece

Key Fact: Match the pieces, not the outline. Wrong figures are often built to look complete and tidy.