AKA “What the heck do these lines mean???”
Top front end (TFE) is probably the hardest section to initially understand in the PAT, but fear not! Once you get a hang of it and what all the different lines mean it becomes an easy section to master. Take your time to read through this tutorial, as slow as you need to, re-read it, and refer back to it when you need to clarify something.
Before we get started, we need to get you caught up on all the terminology, and explain what the problem is really asking you to do. By the end of this tutorial you should be able to answer some easy TFE questions, and with practice, you’ll be able to answer the hard ones as well. Let’s get started!
Objective of TFE Section: Take the two 2D views, imagine the 3D shape they represent, and find the missing 2D view. It’s OK if that doesn’t make sense yet, just keep reading and keep this in the back of your mind.
Explaining What Each View Means
Still learning the basics
Note: You will NOT be given the 3D object for real TFE questions, your job will be to imagine the 3D object from the 2D views. But, for the purpose of the tutorial, we’re going to show and use the 3D object to help ease you into solving TFE problems.
Imagine for the box pictured below, I asked you to draw me what the box would look like if you were looking from the top straight down, like so:
It would just look like a rectangle, as drawn below. This is known as the TOP view of the object.
What about the front view? Imagine what the box would look like if you were looking at it directly from the front, like this:
It would look similar to the top view, just a little thinner since the front is thinner than the top. This is known as the FRONT view of the object.
Finally, how would this object look from the right side of the object, like so:
It would look like the front view, only a little shorter this time. It would still retain the same height, since both the front and the right side share a common edge and are the same height in the 3D object. This right side view is also known as the END view. Whenever you see the END view, it’s synonymous with the right side view, remember that.
Explaining What Solid and Dotted Lines Mean
We’re starting to get there…
Now we’re going to do the same thing, but this time our object is a little different. It has a hole in the middle of it. Let’s see how this affects our drawing of the box.
Starting with the TOP view, imagine what this box would look like if you were looking straight down at it from the top, as shown below:
It would look like a smaller rectangle placed within a larger rectangle. Notice we only see the highlighted edges of the shape from the TOP view.
Excellent. Now let’s move to the Front and End views. I want you to do the same thing, just draw what you directly see if you were looking at the object from the Front and End view, as shown below:
All we would see is a flat rectangular wall, similar to what we had in the previous problem with a solid box.
Wait a minute… but how can this object, which has a hole in the middle of it, have the exact same FRONT and END views as the previous solid object?
These two objects are fundamentally different. One of them has a giant hole in the middle of it. There has to be a way to tell them apart based on their views. Let’s look at the x-ray view of the shape with a hole in it.
If we look at the Front view now, what’s fundamentally different compared to the original solid box is the presence of the highlighted hidden edges in the object. There is a sharp change in the depth of the object on the highlighted edges, which we cannot see from the Front or End view in this case.
So for the edges of the object we could directly see, we used solid lines. For these new hidden edges we cannot directly see, because they are behind a solid piece of the object, we use dotted lines. Let’s apply that to the front and end views of our diagram.
To recap, the edges of the object we can directly see from our point of view are drawn as solid lines. The hidden edges of the object – sharp changes in the depth of the object that we CANNOT directly see from our point of view – are drawn as dotted lines.
Superimposition of Solid and Dotted Lines
Here comes the tricky part
OK, we’re going to draw the top, front, and end views one more time, this time for a more complex shape.
Try drawing the Top view of this object yourself, imagine what it would look like:
Great, now let’s do the End view next. Remember, everything you can directly see will be drawn as a solid line, but the hidden edges you CANNOT see are drawn as dotted lines!
As you see from the End View, we see the solid blue lines, which is the outline of the shape, along with 3 green dotted vertical lines, which represent the hidden edges, or changes in depth we cannot see.
Now we’re going to move to the Front view of the object. Try it yourself, think about what it would look like, and continue reading. Be careful! Try to figure out what will be a solid line and dotted line (or if there are any dotted lines at all?)
Answer: Sure, those hidden edges are there, but they are superimposed (or directly covered) by the blue solid lines here:
These two directly visible edges from the Front view:
Are positioned directly over the two hidden edges, therefore you’ll only see solid lines here in the Front view.
So if there is a solid line on top of a dotted line in a particular point of view, the solid line will win out.
Quick Lesson on Orientation
Learn how the different views relate to each other
Now that we’re familiar with the terminology and what the solid and dotted lines mean, I’m going to show you how each view relates with each other, and how you can find a unique feature in the 3D shape in all three views.
Remember, on the real TFE section, you won’t be given a 3D view of the model. You have to imagine the 3D model in your head, and figure out what the missing 2D view would look like. Sounds complicated, but once you get the hang of it, it gets much easier.
Relating the TOP and FRONT Views
Making sense of the views
Let’s start off by relating the Top and Front views. These are the easiest views to relate, because they’re already stacked on top of each other. This means the Left and Right sides of the top and front views line up, as shown:
Therefore, if I wanted to know how the red dot in the Top view related to the Front view, I could just look for a feature in the same position.
Bonus lesson: If you only saw the red dot in the Top view, and were looking for that feature in the Front view, would you expect to find a solid or dotted line?
Well, if you were looking from the Front view at the Top view, you would directly see that sharp edge in that red dot, so you would expect to see a SOLID line from the Front view to represent this directly visible edge. This is the skill we will eventually develop to help beat this section.
Relating the FRONT and END Views
Making sense of the views
Relating the Front and End views is pretty easy too. The top of the Front view is the same as the top of the End view. The bottom of the Front view is the bottom of the End view. Easy enough.
Relating the TOP and END Views
Making sense of the views
OK, this one isn’t as easy at first, but there is a trick to it.
All you have to do is rotate the Top View 90 degrees clockwise to have it stacked on top of the End View, allowing you to compare and relate the two views.
With the Top and End Views stacked on top of each other, you can easily see how each feature in the Top View corresponds to the End View. Notice how all the features highlighted in red in the Top View corresponded with the lines in the End View.
Now on the real DAT, you won’t be able to flip the Top View 90 degrees to make it easy, this was just to show you how the two views relate with each other.
So how would you figure out how the feature highlighted in blue would match up with the End View?
Well, you know the blue dot is close to the back of the shape, since it’s close to the yellow line. Going from Back to Front in the Top View, it’s also the first feature seen. Therefore, going from Back to Front in the End View, it should be just to the left of the Back yellow line, as shown.
Summary of Relating the Top, Front, and End Views
Try it yourself! Here's a practice TFE problem.
Now that we’ve covered the basics and terminology, we’re ready to start solving an example question. Remember, you’re expected to take the information in the two given 2D views, construct a 3D image in your head, and then figure out what the remaining 2D view looks like. Graphically pictured below:
Our Strategy to Answering TFE Questions
Use this algorithm when answering TFE questions
Focus on eliminating the wrong answer choices, don’t look for the correct one. Follow this algorithm:
- Focus on one unique feature of the shape (figuring out which feature to focus on improves with experience and more practice).
- Compare the two given views to figure out how that unique feature looks, try to form a 3D shape in your mind.
- Figure out how the unique feature would look like in the missing view.
- Look at the answer choices, and eliminate the incorrect answer choices.
- Repeat until only the correct answer remains.
So starting with the Front View, let’s figure out how these two highlighted edges in the shape would match up with the End view. These two edges are kind of unique in the Front view and a good place to start.
Recall that the top of the Front View is the same as the top of the End View, and the same for the bottom of the shape.
Therefore, this teal dot:
Matches up this dotted line in the End View. (The same can be said for the purple dot.)
Note the reason these lines are dotted is because of the height of this wall:
Looking from the End View at the Front View, the height of this wall would block our ability to see these two highlighted features, therefore they would be dotted in the End View, as we have observed.
Now, the big question, how would you expect these two highlighted lines to look from the Top View (the missing view) at the Front View?
Notice that from the Top View looking at the Front View, you can directly see the two highlighted edges. They are unobstructured, and run through the entire depth of the shape. Therefore, from the Top View, you should expect to see two solid lines running through the entire length of the Top view at the highlighted positions. Let’s check out the answer choices.
We can see answer choices [A], [B], and [D] match up with what we expected. Answer choice [C] says those two lines would be dotted, or hidden. This cannot be true, since we saw if we looked straight down from the Top View at the Front View, we could directly see those edges. Eliminate [C].
So we have identified a unique feature, figured out how it should look in the missing view, and eliminated any answer choices that didn’t match up. Now let’s focus on another feature. Look at the piece highlighted in red.
Remember your relative positioning. In which part of the shape would you expect to find this feature? Top, bottom, left, right, etc?
We can see that the highlighted feature is located in the bottom–right of the shape.
Notice the feature is made of solid lines, which means we can directly see this feature from the Front view. Also, due to its positioning and the End View, we can tell that the feature is a little rectangular block sticking out of the shape.
So here’s the big question again, what would this feature look like in the Top View?
To quickly recap, we know that this feature is a little rectangular piece sticking out of the shape located in the bottom–right. It is also located on the Front of the shape.
Answer [B] can be immediately eliminated because it places the feature on the left side of the shape, not the right.
Now we’re left with just [A] and [D]. TIP: Once you’re down to two answer choices, you can just compare what’s different about them, and eliminate the wrong answer choice. The difference between these two choices is this line:
Looking back at the original question, notice that the feature is on the bottom of the shape. Therefore, if you were looking from the Top View at the Front View, you would see a solid line showing the sharp break in the shape. Answer [A] is basically saying the feature in question is attached to the top of the object, not the bottom. In other words, answer [A] says the feature is flush with the top of the shape.
The line highlighted in orange can be seen here:
Answer choice [A] could be correct if the 3D shape looked like this. But that would totally change how the Front view and End view look too.
Thus, you can eliminate Answer [A], leaving Answer [D] as the correct answer.
Time to try it yourself
That’s a quick overview of how to tackle the TFE section. You should now have enough information to try a basic problem. If you’re still unsure of some stuff, don’t worry, that’s totally normal. With practice, you’ll begin to fill in the gaps.
One last tip: If you’re having trouble figuring out how a specific feature looks, try to focus on another feature. There are many ways to get the right answer in the TFE section, and you don’t have to understand all of the features to get the right answer. Try to answer this question, take your time, and when you’re ready scroll down to see the answer.
Think you got it? Scroll down to reveal the correct answer with an explanation!
Let’s start with the back-right of the object, with the feature highlighted in teal.
This feature in the Top View matches up with this piece in the Front View.
Because this feature is a rectangle in the Top view, and corresponds to the highlighted part in the Front view, we know it’s a block that is sticking up out of the shape. So we know this piece is a “step” feature in the back-right side of the shape. We know it’s in the back-right because the feature is towards the back of the object in the Top view, and on the right side in both the Top view and Front view.
Looking from the End View at the Top View then, we cannot directly see this edge highlighted in purple, therefore we will see a horizontal dotted line in the middle of the object towards the back half of the shape.
We can see the dotted line we described in Answers [A] and [C]. However, Answers [B] and [D] show a solid line, therefore they can be eliminated.
Now that we have two answer choices left, we can focus on what is different between them. Let’s focus on the highlighted difference in red. You can focus on other features, but they may be harder to figure out. If you’re having trouble figuring out how a feature looks, try looking at another feature. You don’t need to figure out how the entire shape looks to get the right answer, you just need to eliminate the wrong answers.
Answer [C] says the upward protruding block is flush with the right side of the object, since there is no horizontal solid line like in Answer [A]. However, looking at our question, the upward protruding block is on the left side of the shape, not the right side. Therefore, Answer [C] can be eliminated, leaving [A] as the correct answer.
From here it just takes practice.
Don’t worry too much if you don’t immediately understand the TFE section. For many people, they will struggle with this section a lot, and then after a lot of practice and time spent reading and understanding the solutions, everything will suddenly ‘click’ and this will become an easy section. The examples you read may make this section look complicated and time consuming, but I promise it’s not. TFE questions can be solved in less than a minute. You just follow the algorithm at the top of this tutorial: find a unique feature, compare the views, eliminate the wrong answer choices, and repeat.
I recommend practicing with the TFE Visualizer next. This will give you an unlimited amount of problems to practice. Really try to focus and understand what the solid/dotted lines mean, and how to synthesize that information into what the 3D shape will look like. It will take practice, but if you work hard, read the solutions in the PAT exams, and practice with the TFE Visualizer, you’ll come to find this section is easier than you think!
Did you find this tutorial helpful? Let me know!