👻Ghostly Measurements👻
One of the most classic Halloween costumes is the ghost because It’s one of the
easiest costumes to make and all you need is a sheet! Let’s see if we can figure out
how to design the perfect ghost costume.
1. To ensure the sheet hangs evenly all around, we need to cut the sheet into a circle.
How can we determine the size of the circle to cut? Think about the measurements
needed for the circle and how they relate to the trick-or-treater. Draw a quick
sketch to illustrate the measurements.
Answers may vary but students should be using reasonable numbers to decide the measurements.
To determine the size of the circle, students should
relate the person's height to the radius. We should
subtract 10 to 12 inches (~30cm) from the person’s
height to ensure the sheet doesn’t drag on the ground
and cause tripping.
Also, the size of the person's head will affect how low
the sheet hangs. Since the hat fits over the head, we
need to add at least 2 inches to the radius of the circle
(so 4 inches in diameter) to account for the head's
diameter.
2. To make sure the costume is safe and spooky, it should hang about 12 inches
above the ground. Using the person’s height, write a formula to calculate the
diameter for the entire circle (i.e. entire costume).
Answers may vary but students should be using reasonable numbers.
We make adjustments to ensure the sheet properly hangs over the head and doesn’t drag on the floor.
𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑠𝑡𝑢𝑚𝑒 = 2 × (𝑝𝑒𝑟𝑠𝑜𝑛'𝑠 ℎ𝑒𝑖𝑔ℎ𝑡 − 12 𝑖𝑛𝑐ℎ𝑒𝑠 + 2 𝑖𝑛𝑐ℎ𝑒𝑠)
Here, the radius represents the distance from the person’s head (with an additional 2 inches for the head) to just
above the ground (12 inches above, to prevent tripping). Since the full circle covers both the front and back of the
person, we multiply the radius by 2 to calculate the total diameter of the circle.
Here is a table of commercially available sheet sizes:
Sheet size
Inches
Centimeters
Twin
66 x 96
167 x 243
Long Twin
66 x 102
167 x 259
Full
81 x 96
205 x 243
Queen
90 x 102
228 x 259
King
108 x 102
274 x 259
3. Now, let’s design a ghost costume for yourself! Using your height, find which
commercially available sheet size will give you the best fit with the least amount of
waste.
Answers will vary.
Sample solution: The height of the trick-or-treater is 161cm.
1 𝑖𝑛𝑐ℎ
𝐻𝑒𝑖𝑔ℎ𝑡 𝑖𝑛 𝑖𝑛𝑐ℎ𝑒𝑠 = 161𝑐𝑚 × 2.54 𝑐𝑚 = 63. 39 𝑖𝑛𝑐ℎ𝑒𝑠
𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑠𝑡𝑢𝑚𝑒 = 2 × (𝑝𝑒𝑟𝑠𝑜𝑛'𝑠 ℎ𝑒𝑖𝑔ℎ𝑡 − 12 𝑖𝑛𝑐ℎ𝑒𝑠 + 2 𝑖𝑛𝑐ℎ𝑒𝑠)
𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑠𝑡𝑢𝑚𝑒 = 2 × (63. 39 𝑖𝑛𝑐ℎ𝑒𝑠 − 12 𝑖𝑛𝑐ℎ𝑒𝑠 + 2 𝑖𝑛𝑐ℎ𝑒𝑠)
= 106. 78 ≈ 107 𝑖𝑛𝑐ℎ𝑒𝑠
The largest sheet available is the King size (108 x 102 inches), but the largest circle you can cut from it would have a
diameter of 102 inches, which is smaller than the 107 inches you need. This means that no commercially available
sheet will perfectly fit your height without leaving extra fabric or requiring more than one sheet!
4. Let’s explore this further: For each commercially available sheet size, determine the
maximum height of a person who can make a perfectly circular ghost costume
using only one sheet.
For each sheet size, the diameter needs to be less than or equal to the smaller dimension of the sheet (because
that limits the largest possible circle that fits within the rectangle).
for Twin & Long Twin sheets:
2 × (𝑝𝑒𝑟𝑠𝑜𝑛'𝑠 ℎ𝑒𝑖𝑔ℎ𝑡 − 12 𝑖𝑛𝑐ℎ𝑒𝑠 + 2 𝑖𝑛𝑐ℎ𝑒𝑠) ≤ 66 then solve for the person’s height by simplifying the
inequality
(𝑝𝑒𝑟𝑠𝑜𝑛'𝑠 ℎ𝑒𝑖𝑔ℎ𝑡 − 10 𝑖𝑛𝑐ℎ𝑒𝑠) ≤ 33
𝑝𝑒𝑟𝑠𝑜𝑛'𝑠 ℎ𝑒𝑖𝑔ℎ𝑡 ≤ 43
𝐻𝑒𝑖𝑔ℎ𝑡 (𝑐𝑚) = 43 𝑖𝑛𝑐ℎ𝑒𝑠 ×
2.54 𝑐𝑚
1 𝑖𝑛𝑐ℎ
≈ 109. 2 𝑐𝑚 (3 feet 7 inches); This is around the average height of 5 year
olds.
Sheet size
Inequality
Maximum
Person’s Height
Twin & Long twin
2 × (𝑝𝑒𝑟𝑠𝑜𝑛'𝑠 ℎ𝑒𝑖𝑔ℎ𝑡 − 12 𝑖𝑛𝑐ℎ𝑒𝑠 + 2 𝑖𝑛𝑐ℎ𝑒𝑠) ≤ 66
43 inches
≈109.2 cm
Full
2 × (𝑝𝑒𝑟𝑠𝑜𝑛'𝑠 ℎ𝑒𝑖𝑔ℎ𝑡 − 12 𝑖𝑛𝑐ℎ𝑒𝑠 + 2 𝑖𝑛𝑐ℎ𝑒𝑠) ≤ 81
50.5 inches
≈128.2 cm
Queen
2 × (𝑝𝑒𝑟𝑠𝑜𝑛'𝑠 ℎ𝑒𝑖𝑔ℎ𝑡 − 12 𝑖𝑛𝑐ℎ𝑒𝑠 + 2 𝑖𝑛𝑐ℎ𝑒𝑠) ≤ 90
55 inches
≈139.7 cm
King
2 × (𝑝𝑒𝑟𝑠𝑜𝑛'𝑠 ℎ𝑒𝑖𝑔ℎ𝑡 − 12 𝑖𝑛𝑐ℎ𝑒𝑠 + 2 𝑖𝑛𝑐ℎ𝑒𝑠) ≤ 102
61 inches
≈154.9 cm
5. What do you notice about the maximum heights you calculated for each sheet size?
They can only accommodate certain age groups (most likely up to middle school students in general) but may not
be appropriate for taller children or adults as the maximum height for each sheets only reach up to approximately
155 cm.
6. Analyze the following graph that represents the relationship between sheet
dimensions and the maximum height of a person for a ghost costume. Discuss or
write as much as possible about this graph.
Hints: What do the x and y -axis represent in the context of this graph?
Why is this graph shaded? What does that mean? What are these coordinates shown
on the graph?
The x-axis represents the dimensions (specifically the width or the shorter side) of the commercially available sheet
sizes and the y-axis represents the height in inches.
The shaded region or part below the line emphasizes that any person shorter than the maximum height can also
use that sheet size. The line represents the general inequality:
Let 𝑥= minimum dimension of each bedsheet and y= person’s height
Then 2 × (𝑦 − 12 𝑖𝑛𝑐ℎ𝑒𝑠 + 2 𝑖𝑛𝑐ℎ𝑒𝑠) ≤ 𝑥 simplifies to
𝑦≤
𝑥
2
+ 10
Each point, such as (66, 43), (81, 50.5), (90, 55), and (102, 61), shows the exact sheet size and the maximum height
of a person who can use that sheet.
For example:
A twin-sized sheet (66 inches wide) can accommodate a person up to 43 inches tall.
These points mark the cutoffs—beyond these heights, the sheet won’t be large enough to make the costume.
Teaching notes:
Encourage further discussion! “If you are taller than 61 inches (154.9cm), what would you do?”
“Looking at the graph, the y intercept is 10 inches, what does this mean in this context?”
For Question 6, instead of interpreting or labeling the inequality from the graph, you could modify the question to
have students draw the inequality themselves based on the relationship shown.
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