Spatial Reasoning Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Spatial Reasoning.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
The cognitive ability to visualize, manipulate, and reason about two- and three-dimensional objects mentally in space.
Imagining how furniture will fit in a room before physically moving any of it.
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: Spatial reasoning is mentally rotating, flipping, and fitting shapes without touching them.
Common stuck point: The procedure for spatial reasoning is the easy part; the trap is assuming a turned or flipped shape is a new shape. Asking "Am I asked to mentally move or fit shapes, not to measure or compute a number?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint: Ask: Am I asked to mentally move or fit shapes, not to measure or compute a number?
Worked Examples
Example 1
easyLook at this sequence of shapes: triangle, square, pentagon. How many sides does the next shape have? Draw a rough sketch.
Answer
A hexagon (6 sides).
First step
1
Step 1: Count the sides: triangle sides, square sides, pentagon sides.
Full solution
- 2 Step 2: The pattern increases by side each time:
- 3 Step 3: The next shape has sides — a hexagon.
Spatial reasoning involves recognising patterns in shapes. Here the number of sides increases by one each step, a simple arithmetic sequence. Visualising the shapes mentally helps predict the next one.
Example 2
mediumA cube is painted red on all 6 faces and then cut into smaller equal cubes. How many small cubes have paint on exactly 2 faces?
Example 3
easyA puppy hides under the bed. Is the puppy on or under the bed?
Example 4
easyA fish swims outside the bowl. Wait, that's wrong! A fish swims inside the bowl. Is the fish in or out?
Example 5
mediumToys are inside the toy box. Crayons are outside the toy box on the floor. Which toys are in the box?
Example 6
easyThe school is right next to your house. A park is far away on the other side of town. Which one is near your house?
Example 7
easySam stands behind Ben in line. Is Ben in front of or behind Sam?
Example 8
mediumLily stands in front of the mirror. Her reflection raises its right hand when Lily raises her left hand. Where is Lily's left hand: on the side near her heart or the other side?
Example 9
easyThe letter is flipped over a horizontal line (top to bottom flip). Which letter does it look like?
Example 10
easyA red ball sits in front of you and to your left. Describe its position with two words: ___ and ___.
Example 11
mediumA rectangle is standing tall (long side up-down). You turn it degrees. Is the rectangle now tall or wide?
Example 12
easyYou cut a square into 4 equal parts with one cut down the middle and one cut across the middle. Each piece is called a what?
Example 13
easyOn a simple map, north is up and south is down. You walk north from the school to the park. To return to the school, which direction do you walk?
Example 14
mediumSam folds a square paper in half. Then she cuts a small half-circle on the folded edge. When she opens the paper, what shape did she cut out?
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easyIf you fold a square piece of paper in half and then in half again (both folds along the middle), how many layers of paper are there when unfolded? How many crease lines are visible?
Example 2
mediumA rectangular box is cm long, cm wide, and cm tall. What is the length of the space diagonal (corner to opposite corner)?
Example 3
easyA cube has how many faces?
Example 4
easyIf you look at a cylinder (like a can) from directly above, what shape do you see?
Example 5
easyFrom the side, a ball (sphere) looks like what shape?
Example 6
easyHow many edges does a cube have?
Example 7
easyIf you unfold a cube flat, how many square faces are in the flat pattern (net)?
Example 8
easyA shape looks like a triangle from the front and a circle from the top. What 3D object could it be?
Example 9
easyHow many corners (vertices) does a cube have?
Example 10
easyYou rotate the letter 'b' upside down (180°). Which letter does it look like?
Example 11
mediumA net has 6 squares in a cross shape. When folded, what solid does it form?
Example 12
mediumA solid is built from 4 unit cubes stacked in an L-shape. How many cubes touch the ground if it sits on its longest side?
Example 13
mediumA cube is built from unit cubes. How many small cubes are there?
Example 14
mediumLooking at a stack of cubes from the front you see 3, from the side you see 2, and it's 1 cube tall. What is the maximum number of cubes?
Example 15
mediumA square piece of paper is folded in half, then in half again. How many layers thick is it now?
Example 16
mediumWhy might a single 2D drawing of a 3D object be misleading?
Example 17
mediumA cube is painted red, then cut into 8 small equal cubes (). How many small cubes have exactly 3 red faces?
Example 18
mediumTwo shapes look identical but one is shown rotated. What spatial skill confirms they're the same shape?
Example 19
challengeA cube is painted and cut into 27 unit cubes. How many have NO paint at all?
Example 20
challengeOn a cube net (cross shape), two squares are on opposite ends of the long arm. When folded into a cube, are they adjacent or opposite faces?
Example 21
challengeA solid looks like a square from the front, a square from the side, and a circle from the top. What single solid fits all three views?
Example 22
challengeA painted cube is cut into 27 unit cubes. How many have exactly 2 painted faces?
Example 23
easyA bird sits on top of a tree. A bunny sits on the ground. Is the bird above or below the bunny?
Example 24
easyA cat is inside the box. Is the cat in or out of the box?
Example 25
easyA book sits on a table. Is the book on or under the table?
Example 26
easyA balloon floats above your head. Is the balloon above or below you?
Example 27
mediumYour shoes are on the floor below your bed. Are the shoes above or below the bed?
Example 28
hardA sun is in the sky. A flower is on the ground. Is the sun above or below the flower?
Example 29
easyYou hold up your right hand. Which hand is on the other side, your left or right?
Example 30
easyMaya stands in line. The teacher is in front of her. Where is the teacher: in front of or behind Maya?
Example 31
easyMost people write with their right hand. The hand you do NOT use to write is usually the what?
Example 32
mediumA star is very far up in the sky. The moon is closer to Earth than the star. Which one is near to Earth: the star or the moon?
Example 33
mediumA rectangle has how many corners?
Example 34
easyYou turn a square degrees (a quarter turn) to the right. What shape do you still see?
Example 35
easyA triangle points UP. You turn it degrees to the right (a quarter turn). Which direction does the point now face?
Example 36
easyAn arrow points right. You flip it over a vertical line (left-right flip). Which direction does it now point?
Example 37
mediumA triangle points UP. You turn it degrees to the left (a quarter turn left). Which direction does the point now face?
Example 38
mediumA dog is behind the chair and to the right of the chair. Where is the dog? Use both position words.
Example 39
easyYou put two squares of the same size right next to each other so their sides touch. What new shape do they make?
Example 40
easyYou cut a square into 2 equal parts straight down the middle. Each part is called a what of the square?
Example 41
easyYou put two equal triangles together so a long side of each touches. They can form a square. How many triangles make the square?
Example 42
mediumYou take 6 equal triangles (the kind with one short side and one slanted side meeting at a corner) and arrange them around a center. What shape do they form?
Example 43
mediumOn a map of the schoolyard, the slide is east of the swings. From the swings, which direction is the slide: left or right (if north is up at the top of the map)?