🧱 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

🧱 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.

🧱 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.

🧱 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?

🧱 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.

🧱 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

🧱 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.