Where Is the Missing Area? Two Puzzles to Challenge Your Thinking.

I am a school teacher with a love for writing short stories, usually with a humorous twist.

It’s Obvious Once You Know The Answer!

You come across a riddle, stew over the answer and then frustratingly give up. How many times has that happened to you?

After looking at the answer and exclaiming, ‘Of course, that’s obvious', what follows is a tirade of curses and self-chastisement for not noticing the ‘obvious’. Finally, you vow that ignorance will never be your partner again; at least until it happens again next time.

Keep this in mind when you attempt the two puzzles described below. They are especially suited for anyone who has a basic knowledge of trigonometry involving triangles. However, all required information is provided and solutions can be established on basic levels. Hence, all lovers of a challenge are invited to enter this arena of geometrical paradoxes.

P.S. Resist the urge to jump to the solution until you have sweated over your efforts for a sufficiently long period of time!

Puzzle 1

The rectangle shape below consists of 65 small squares. There are three right-angled triangles (shape 1, shape 2 and shape 3) and two other shapes (shape 4 and shape 5).

We expect that the sum of the areas of the five shapes should be 65, but let’s check this.

To calculate the area of a triangle we use:

For shape 1, the base is 13 and the height is 5, so its area is 0.5 × 13 × 5 = 32.5 squares.

Similarly, shape 2 has area 0.5 × 5 × 2 = 5 squares and shape 3 has area 0.5 × 8 × 3 = 12 squares.

Shape 4 has area 8 squares and shape 5 has area 7 squares.

The total of the areas is 32.5 + 5 + 12 + 8 + 7 = 64.5 squares.

We know that the whole shape has area 65 squares, and yet by finding the sum of the five individual shapes the area is 64.5 squares.

Where is the missing 0.5 square?

The Explanation

In fact, the exact area of the whole shape is 64.5 squares. There is no missing area. If the five shapes are separately drawn and arranged to form the large rectangle, you will notice that they do not fit exactly (shown below). There is space between some of the shapes whose total area is 0.5 squares, which explains the mythical 0.5 squares that is supposed to be missing. You may like to try drawing the shapes to convince yourself.

One proof is to show that the shapes do not fit together perfectly because they are not similar shapes.

For a pair of similar shapes, the ratio of their corresponding side lengths are different. This is examined below.

If line AF is to be smooth at point D, then triangle ABD must be similar to triangle DEF. This means the ratio of the lengths of corresponding sides is the same.

Also, triangle DEF must be similar to triangle ACF.

Finally, triangle ABD must be similar to triangle ACF.

Since the ratios are not the same, the hypotenuses of the triangles do not join perfectly, leaving a gap.

Proof Using Angles

Basic trigonometry can be used to find angles in a right-angled triangle when the length of two sides is given.

To determine each angle, we will use the trigonometric inverse tangent function found on a scientific calculator.

If line DF is to be part of line AF, then:

* angle a is the same as angle e

* angle b is the same as angle f

* angle c is the same as angle e

* angle d is the same as angle f

Each pair of angles would be alternate angles if they had the same magnitude. For example, angle c is 21.800 and angle e is 21.040, which are not the same.

A requirement for there to be no gaps in the shape is that all pairs of angles we have found are alternate angles. This is not true, so the five shapes do not fit together properly.

Now, onto the second puzzle!

Puzzle 2

For this puzzle, two triangles of the same size and two trapezia of the same size are put together to form a rectangle shape comprising 65 small squares.

The four shapes are rearranged to form the shape below, which consists of 64 small squares.

Where is the missing square?

As in puzzle 1, the exact area is in fact 64 squares. To show this, find the area of each of the four shapes and show that their sum is 64.

Note: To find the area of a trapezium, use:

To explain the missing square, use the following guide.

• Draw the four shapes accurately. You will see that there is a gap between the shapes whose area is 1 square.
• Use the discussion given in Puzzle 1 to show that angles that should be the same are actually not the same.

Challenge

The area of the gap has the shape shown below.

If you want to test your mathematical ability, provide more detailed calculations to the method described below.

Steps

1. Apply Pythagoras’ theorem, a2 + b2 = c2 to triangle ABC and to triangle AEF to show that:

2. Use the inverse tangent function on your calculator to find the following angles.

3. Find the area of triangle ACE using area = 0.5 × a × b × sin(q), where a and b are the lengths of the two sides of the triangle and q is the angle formed by a and b.

4. The area of triangle ACE is the same as triangle CED.

Hence, the area of the gap is 2 × 0.4992 = 0.9984

Allowing for rounding errors when working out square roots, the area of the gap is 1.