Choose "One frame" or "A row of identical frames", then enter the frame size, the hanger position and the target height in cm. If the target center height is blank, the eye-level guideline of 145 cm is used.
Frame size
cm
cm
Hanger
cm
Target height
cm
The result is the height of the point where the hanger rests on the hook. Depending on the hook's shape, the nail goes a few millimeters higher or lower, so hold the real hook against the wall to check.
Result and figure
Enter the frame height, the hanger position and the target height on the left and press "Calculate". The nail height, the heights of the top and bottom of the frame (and each nail position for a row) will appear here with a front view of the wall.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and figures
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Symbols and terms
-
Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
-
NumberChef Calculators List
What you can do on this page
- From the frame height, the hanger position (the distance from the top of the frame to the hanger or the peak of the wire) and the target center height (by default eye level, 57 in from the floor), find the height for the nail or hook and the heights of the top and bottom of the frame
- Above a sofa or a TV stand, set the "bottom of the frame" height (top of the furniture + a gap) instead, and the center, top and nail heights are worked out from it
- For a frame hung by a wire, find how high the peak of the wire rises when pulled tight from the D-ring position, the D-ring spacing and the wire length (with the Pythagorean theorem), and get a nail height that allows for the slack
- To hang identical frames in a row, enter the wall width, the number of frames and the frame width, then choose "set the gap and center" or "space evenly across the wall" to get the total width, the side margins and the \(x\) position of each nail (from the left end of the wall)
- Check the result on a front view of the wall (floor, eye level, frames, nails). For a frame hung from two hangers, the positions of both nails are shown
"Center the art 57 in above the floor" (the 57-inch rule) is an eye-level guideline used by museums and galleries; some use 60 in. How it looks depends on the ceiling height, the furniture and the people looking, so change the target height as you like. The nail height depends on where the hanger is on the frame, so always measure the real hanger position (for a wire, to the peak when pulled up tight) first. Switch "Units" above the calculator to Metric to work in centimeters.
What is this calculation used for?
Hanging art or a poster above the living room sofa
If the top of the sofa back is 34 in above the floor, putting the bottom of the frame 6 to 10 in above it (40 to 44 in) makes the sofa and the art look like one group. For a 24 in tall frame with its bottom at 42 in, the center is at 54 in and the top at 66 in; with the hanger 3 in below the top, the nail goes at 63 in.
Whether eye level (57 in) or the gap to the furniture comes first depends on the room. In a room where people mostly sit, favor the gap to the furniture, even if the center ends up a little lower.
A row of family photos in a hallway or on a stairway wall (gallery wall)
Three 16 in wide frames with 2 in gaps on a 120 in hallway wall have a total width of 52 in and side margins of 34 in, with frame centers 42, 60 and 78 in from the left end. Write these positions on paper before marking the wall, and you will not end up with extra holes from rehanging.
Even spacing on the same wall gives wide 18 in gaps that can look scattered, so for smaller frames like photos, setting the gap and centering the group makes it read as one piece.
Keeping holes to a minimum in a rental (moving or redecorating)
In a rental, every rehang adds holes, so working out the nail height and position first and getting it right the first time pays off. In drywall, picture hooks work and leave holes about the size of a thumbtack.
The calculated height is where the hanger rests on the hook, so the nail itself goes about 1/8 in higher or lower depending on the hook. Hold the hook against the wall so its resting point is at the calculated height, and you will get it right the first time.
A wide frame or canvas on the wall above the TV
If the top of the TV screen is 45 in above the floor, putting the bottom of the frame about 6 in above it (51 in) keeps the TV and the art neither too close nor too far apart. For a 12 in tall frame, the center is at 57 in, right at the eye-level guideline.
To line frames up across the width of the TV, enter the TV width as the wall width, and the frames will be placed evenly on both sides of the TV's center. The screen width and height come from the screen size in inches (the diagonal) and the aspect ratio.
Deciding the nail position for a heavy picture or mirror on a wire
On a frame with a wire, the slack moves the nail height up. A 20 in wire between D-rings 8 in below the top and 16 in apart rises 6 in above the D-rings, so the peak lands 2 in below the top of the frame. Subtract that 2 in to set the nail height.
If the wire is too long, the peak rises above the top of the frame and the wire shows on the wall. Keep the wire just a little longer than the D-ring spacing, and for heavy frames, use two hooks to share the load or screw into a stud.
Lining up children's artwork or certificates at the same height
Four certificate frames for 8.5 × 11 in paper (about 12.5 in wide and 10 in tall outside, hung sideways) spaced evenly on a 72 in wall have gaps and margins of about 4.4 in, with frame centers starting about 10.65 in from the left end and every 16.9 in after that. On a wall for children, a lower center height (about 48 to 51 in) is easier for them to see.
When you add more frames, just recalculate the spacing with the same formula to rehang them on the same wall.
Hanging works at a consistent height in exhibitions, galleries and offices
Museums and galleries widely line up the center height of works at about 57 in above the floor, whatever their size. Measure each work's height \(H\) and hanger position \(w\), and \(h = 57 + H \div 2 - w\) gives the nail height, so large and small works share the same eye line.
If the checklist of works includes the frame size and hanger position, this formula alone is enough to lay out the hanging positions for the show.
Formulas and figures
Top and bottom of the frame and the nail height (one frame)
Figure
Standard notation
(the usual math form)
\(T\)
\(=\)
\(c\)
\(+\)
\(H\)
\(\div\)
\(2\)
\(B\)
\(=\)
\(c\)
\(-\)
\(H\)
\(\div\)
\(2\)
\(h\)
\(=\)
\(T\)
\(-\)
\(w\)
In words
(symbols replaced with words)
③
\(T\): top of frame
\(=\)
①
\(c\): center height
\(+\)
②
\(H\): frame height
\(\div\)
\(2\)
④
\(B\): bottom of frame
\(=\)
\(c\): center height
\(-\)
\(H\): frame height
\(\div\)
\(2\)
⑥
\(h\): nail height
\(=\)
\(T\): top of frame
\(-\)
⑤
\(w\): top to hanging point
The formula in words
①
Take the
\(c\): center height
②
and add half the
\(H\): frame height
③
to get the
\(T\): top of frame
④
Subtract half the frame height from the center height instead to get the
\(B\): bottom of frame
⑤
From the top of the frame, subtract the
\(w\): distance from the top to the hanging point
⑥
to get the
\(h\): nail height
(put the nail or hook here)
Quick example
Hang a 24 in tall frame so its center is 57 in above the floor (eye level). The sawtooth hanger is 2 in below the top of the frame.
top of frame \(T\)
\(=\)
center height (57 in)
\(+\)
frame height (24 in)
\(\div\)
\(2\)
nail height \(h\)
\(=\)
top of frame (69 in)
\(-\)
top to hanging point (2 in)
\(T = 57 + 24 \div 2 = 69\ (\mathrm{in}),\quad B = 57 - 12 = 45\ (\mathrm{in})\)
\(h = 69 - 2 = 67\ (\mathrm{in})\)
Key idea
The basic rule for hanging art is to put the center of the frame, not the top, at eye level. Museums and galleries widely use the guideline of centering art about 57 in (145 cm) above the floor, the "57-inch rule" (some use 60 in). It is close to the eye height of a standing person, and using it for large and small frames alike makes the art in a room look evenly placed. In a room with a low ceiling or where people mostly sit, you can go a little lower (53 to 55 in), and in a home of tall people a little higher. Change it freely.
Above a sofa or a TV stand, the gap to the furniture matters more than the center height. Leaving about 6 to 10 in above the sofa back, or about 6 in above the top of a TV screen, makes the furniture and the art look like one group. In that case, set "Bottom of the frame" in the calculator. The center height \(c\) is then the bottom \(B\) plus half the frame height.
The nail height \(h\) is the top of the frame \(T\) minus \(w\), the distance from the top of the frame to the point that rests on the nail. This \(w\) is very different from frame to frame: for a sawtooth hanger or screw eye, it is the distance from the top to the hanger; for a wire, to the peak of the wire pulled up tight. Even for frames you want at the same height, different hanger positions give different nail heights, so measure \(w\) on each frame.
Height of the wire peak when pulled tight (allowing for slack)
Figure
Standard notation
(the usual math form)
\(r\)
\(2\)
\(=\)
\(\dfrac{\ell}{2}\)
\(2\)
\(-\)
\(\dfrac{s}{2}\)
\(2\)
\(w\)
\(=\)
\(d\)
\(-\)
\(r\)
In words
(symbols replaced with words)
③
\(r\): rise of the peak above the D-rings
squared
\(=\)
①
half the wire length \(\ell\)
squared
\(-\)
②
half the D-ring spacing \(s\)
squared
⑤
\(w\): top to hanging point
\(=\)
④
\(d\): top to D-rings
\(-\)
\(r\): rise of the peak above the D-rings
The formula in words
①
Square
half the wire length \(\ell\)
②
and subtract the square of
half the D-ring spacing \(s\)
③
to get the square of the
\(r\): rise of the peak above the D-rings
(take the square root to get \(r\))
④
Subtract \(r\) from the
\(d\): distance from the top to the D-rings
⑤
to get the
\(w\): distance from the top to the hanging point
(use it as \(w\) in the first formula)
Quick example
A 20 in wire is strung between D-rings 8 in below the top of the frame and 16 in apart. When you hang it on a nail and the wire pulls tight, how far below the top of the frame is the peak?
rise of the peak \(r\)
squared
\(=\)
half the wire (10 in)
squared
\(-\)
half the spacing (8 in)
squared
top to hanging point \(w\)
\(=\)
top to D-rings (8 in)
\(-\)
rise of the peak (6 in)
\(r^{2} = 10^{2} - 8^{2} = 100 - 64 = 36,\quad r = \sqrt{36} = 6\ (\mathrm{in})\)
\(w = 8 - 6 = 2\ (\mathrm{in})\)
Key idea
When a frame with a picture wire hangs on one nail, the wire pulls into a V shape and its peak rests on the nail. The peak sits \(r\) above the D-rings holding the wire. As in the figure, the wire from the peak to a D-ring (length \(\ell \div 2\)) is the hypotenuse of a right triangle whose base is half the D-ring spacing (\(s \div 2\)), so the height \(r\) comes from the Pythagorean theorem.
The looser the wire (the longer \(\ell\)), the larger \(r\), so the hanging point moves up and the nail goes higher. If \(r\) is larger than the distance to the D-rings \(d\), then \(w\) is negative: the peak of the wire rises above the top of the frame and the wire shows on the wall. In that case, shorten the wire or move the D-rings a little higher.
More reliable than the calculation is to turn the frame over, pull the wire up tight with a finger and measure \(w\) from the top of the frame to the peak directly. Choose "measure it pulled tight" in this calculator to use that \(w\) as is. The formula from the wire length helps when the frame is not in your hands yet (you only know the wire length from a catalog), or when you want to decide how long to make the wire.
Total width and side margins of a row (set the gap and center)
Figure
Standard notation
(the usual math form)
\(T_{w}\)
\(=\)
\(n\)
\(\times\)
\(a\)
\(+\)
\((\)
\(n\)
\(-\)
\(1\)
\()\)
\(\times\)
\(g\)
\(m\)
\(=\)
\((\)
\(W\)
\(-\)
\(T_{w}\)
\()\)
\(\div\)
\(2\)
In words
(symbols replaced with words)
④
\(T_w\): total width
\(=\)
①
\(n\): number of frames
\(\times\)
②
\(a\): frame width
\(+\)
\((\)
\(n\): number of frames
\(-\)
\(1\)
\()\)
\(\times\)
③
\(g\): gap
⑥
\(m\): side margin
\(=\)
\((\)
⑤
\(W\): wall width
\(-\)
\(T_w\): total width
\()\)
\(\div\)
\(2\)
The formula in words
①
Take the
\(n\): number of frames
②
times the
\(a\): frame width
(\(n \times a\)) and add
③
\(g\): the gap
(\(n - 1\)) times
④
to get the
\(T_w\): total width
⑤
Subtract the total width from the
\(W\): wall width
and divide by 2
⑥
to get the
\(m\): side margin
(the same on the left and right)
Quick example
Hang three 16 in wide frames with 2 in gaps, centered on a 120 in (10 ft) wall.
total width \(T_w\)
\(=\)
frames (3)
\(\times\)
frame width (16 in)
\(+\)
\(2\)
\(\times\)
gap (2 in)
side margin \(m\)
\(=\)
\((\)
wall width (120 in)
\(-\)
total width (52 in)
\()\)
\(\div\)
\(2\)
\(T_{w} = 3 \times 16 + 2 \times 2 = 48 + 4 = 52\ (\mathrm{in})\)
\(m = (120 - 52) \div 2 = 34\ (\mathrm{in})\)
Key idea
To hang identical frames in a row, the basic approach is to decide the gap \(g\) between frames first and center the whole group on the wall. With \(n\) frames there are \(n - 1\) gaps, so the total width is \(n\) frame widths plus \(n - 1\) gaps. For frames of the same size, a gap of about 2 to 3 in is a common guideline; larger frames look balanced with a little more.
The side margin \(m\) is what is left of the wall after the total width, split in half between the left and right. If the margin is very large compared with the gap, the group looks small on the wall, so widen the gap or consider "even spacing" in the next formula. If \(m\) comes out negative (the total width is more than the wall), that many frames with that gap will not fit.
The wall width \(W\) does not have to be the whole wall. Use the width of a sofa or TV stand to hang above it, or the width inside a window or column to avoid it, and the group will be centered on that furniture or space.
Gap for even spacing across the wall (side margins = gaps)
Figure
Standard notation
(the usual math form)
\(g\)
\(=\)
\((\)
\(W\)
\(-\)
\(n\)
\(\times\)
\(a\)
\()\)
\(\div\)
\((\)
\(n\)
\(+\)
\(1\)
\()\)
\(m\)
\(=\)
\(g\)
In words
(symbols replaced with words)
④
\(g\): gap
\(=\)
\((\)
①
\(W\): wall width
\(-\)
②
\(n\): number of frames
\(\times\)
③
\(a\): frame width
\()\)
\(\div\)
\((\)
\(n\): number of frames
\(+\)
\(1\)
\()\)
⑤
\(m\): side margin
\(=\)
\(g\): gap
The formula in words
①
From the
\(W\): wall width
②
subtract the
\(n\): number of frames
③
times the
\(a\): frame width
and divide what is left by the number of spaces (\(n + 1\))
④
to get the
\(g\): gap
⑤
The
\(m\): side margin
is the same value
Quick example
Hang three 16 in wide frames on a 120 in wall so the side margins and the gaps are all the same.
gap \(g\)
\(=\)
\((\)
wall width (120 in)
\(-\)
frames (3)
\(\times\)
frame width (16 in)
\()\)
\(\div\)
\(4\)
\(g = (120 - 3 \times 16) \div (3 + 1) = 72 \div 4 = 18\ (\mathrm{in})\)
Key idea
Even spacing makes the side margins and the gaps between frames all the same within the wall width. With \(n\) frames there are \(n + 1\) spaces in all: the left margin, the gaps between frames (\(n - 1\)) and the right margin. Divide what is left of the wall after the frames into these \(n + 1\) spaces, and the gaps and margins match.
This layout suits filling a hallway wall or the full width of a piece of furniture. But on a wide wall with few frames, the gaps get large and the frames can look scattered. In the example, three frames on a 120 in wall have 18 in gaps, quite wide for 16 in frames. If it feels that way, set a gap of 2 to 3 in with the previous formula and center the group so it reads as one unit.
If the total frame width \(n \times a\) is more than the wall width \(W\), the gap is negative and the frames will not fit. Use fewer frames, smaller frames or two rows.
The \(x\) position of each frame center (nail)
Figure
Standard notation
(the usual math form)
\(x_{i}\)
\(=\)
\(m\)
\(+\)
\(a\)
\(\div\)
\(2\)
\(+\)
\((\)
\(i\)
\(-\)
\(1\)
\()\)
\(\times\)
\((\)
\(a\)
\(+\)
\(g\)
\()\)
In words
(symbols replaced with words)
⑤
\(x_i\): \(x\) position of the center of frame \(i\)
\(=\)
①
\(m\): side margin
\(+\)
②
\(a\): frame width
\(\div\)
\(2\)
\(+\)
\((\)
③
\(i\): frame number
\(-\)
\(1\)
\()\)
\(\times\)
\((\)
\(a\): frame width
\(+\)
④
\(g\): gap
\()\)
The formula in words
①
To the
\(m\): side margin
②
add half the
\(a\): frame width
to get the center of frame 1. From there, move right
③
(the
\(i\): frame number
minus 1) times the frame width plus
④
\(g\): the gap
(\(a + g\) each step)
⑤
to get the
\(x_i\): \(x\) position of the center of frame \(i\)
(for a single center hanger, this is where the nail goes)
Quick example
With a 34 in side margin, 16 in frames and 2 in gaps (the example of the third formula), the centers of frames 1, 2 and 3 from the left end of the wall are
center of frame 2 \(x_2\)
\(=\)
margin (34 in)
\(+\)
frame width (16 in)
\(\div\)
\(2\)
\(+\)
\((\)
frame 2 (2)
\(-\)
\(1\)
\()\)
\(\times\)
\((\)
16 in
\(+\)
2 in
\()\)
\(x_{1} = 34 + 8 = 42\ (\mathrm{in})\)
\(x_{2} = 42 + 1 \times 18 = 60\ (\mathrm{in}),\quad x_{3} = 42 + 2 \times 18 = 78\ (\mathrm{in})\)
Key idea
Mark each frame position as the distance \(x_i\) from the left end of the wall to the frame center, and one tape measure is all you need. The center of frame 1 is the margin \(m\) plus half the frame width, and each frame after that is "frame width + gap" (\(a + g\)) further right. In the example, \(x_2 = 60\) is exactly the middle of the 120 in wall: with an odd number of frames, the middle frame's center lands at the center of the wall.
For frames hung from one center hanger (sawtooth hanger, screw eye or wire), \(x_i\) is the \(x\) position of the nail. For frames hung from two hangers, put two nails, each half the hanger spacing (\(s \div 2\)) left and right of the center. Choose "Two hangers" in this calculator to see both \(x\) positions.
For the height, use the nail height \(h\) from the first formula for every frame. Mark \(x_i\) and \(h\) on the wall, put in the nails, hang the frames, and finish by checking with a level (a phone level app works too) for a neat result. For frames of different sizes, lining up the center height \(c\) and keeping the gaps the same looks natural, but then the \(x\) positions depend on each frame's width, so this formula does not apply as is.
The basic rule is to put the center of the frame at eye level (about 57 in above the floor). The top of the frame is \(c + H \div 2\), and the nail height is the top minus the distance to the hanging point, \(h = c + H \div 2 - w\). For a frame on a wire, \(w\) is the distance to the D-rings \(d\) minus the rise of the peak \(r\), found with the Pythagorean theorem. For a row, either center the total width \(n a + (n-1) g\) on the wall (margin \(m = (W - T_w) \div 2\)) or divide the leftover width into \(n + 1\) equal spaces. Each frame center is at \(x_i = m + a \div 2 + (i-1)(a+g)\).
Symbols and terms
Symbols
| \(c\) | cee | The center height of the frame (from the floor, in), from "center". About 57 in above the floor is a common eye-level guideline and this calculator's default. |
| \(H\) | aitch | The frame height (outside, in), from "height". It is the outside height including the frame; the top and bottom are each \(H \div 2\) from the center. |
| \(T\) | tee | The height of the top of the frame (from the floor, in), from "top". It is found with \(T = c + H \div 2\). |
| \(B\) | bee | The height of the bottom of the frame (from the floor, in), from "bottom". It is found with \(B = c - H \div 2\). Above a sofa or furniture, decide this value first. |
| \(w\) | lowercase double-u | The distance from the top of the frame to the hanging point, where it rests on the nail (in). For a sawtooth hanger, from the top to the hanger; for a wire, to the peak when pulled up tight. |
| \(h\) | lowercase aitch | The nail height (from the floor, in), from "hook". It is found with \(h = T - w\); put the nail or hook here. |
| \(d\) | dee | How far below the top of the frame the D-rings holding the wire are (in), from "depth". It is used to work out \(w\) from the wire length. |
| \(s\) | ess | The spacing between the left and right hangers (in), from "span". For a frame on two hangers, it is also the spacing of the two nails. |
| \(\ell\) | script ell | The wire length (D-ring to D-ring, in), from "length". A cursive \(\ell\) is used so it is not confused with the number 1. |
| \(r\) | ar | How far the peak of the tight wire rises above the D-rings (in), from "rise". It is found with \(r = \sqrt{(\ell \div 2)^{2} - (s \div 2)^{2}}\). |
| \(W\) | capital double-u | The wall width (the space for the row, in), from "wall" or "width". |
| \(n\) | en | The number of frames in the row, from "number". There are \(n - 1\) gaps, and \(n + 1\) spaces with even spacing. |
| \(a\) | ay | The frame width (outside, in). All frames in the row are the same width. |
| \(g\) | gee | The gap between frames (in), from "gap". For frames of the same size, about 2 to 3 in is common. |
| \(T_w\) | tee sub double-u | The total width of the group (in), from "total width": \(n\) frames plus \(n - 1\) gaps. |
| \(m\) | em | The side margin (from the end of the wall to the frame, in), from "margin". When the group is centered, it is the same on both sides. |
| \(x_i\) | ex sub i | The \(x\) position of the center of frame \(i\) (from the left end of the wall, in). The subscript \(i\) is the frame number: \(x_1\), \(x_2\), and so on. |
| \(\sqrt{\ }\) | square root | The square root: the positive number that gives this number when squared. It is used with the Pythagorean theorem to find the height of the wire peak (for example \(\sqrt{36} = 6\)). |
Terms
| Eye level (57-inch rule) | The height of a standing person's eyes. It depends on height, but museums and galleries widely center art about 57 in (145 cm) above the floor; some use 60 in. It is this calculator's default center height. |
| Picture hanger | Any hardware on the back of a frame for hanging it on a nail or hook, such as sawtooth hangers, screw eyes, D-rings and picture wire. The type changes the nail height. |
| Sawtooth hanger | A small toothed metal strip at the top center of the back of a frame. It catches directly on a nail, and the teeth let you shift it a little to level the frame. The distance from the top to the hanger is \(w\). |
| Screw eye | A screw with a ring on the end. Two on the back of a frame hold a wire, or one at the top center hangs directly on a hook. |
| D-ring | A D-shaped ring fixed to the back of a frame on each side. It hangs directly on a hook or holds a picture wire. |
| Picture wire | A wire strung between the left and right D-rings (cord works the same way). On one nail it forms a V, and its peak rests on the nail. The more slack, the higher the peak. |
| Picture hook | A hook held by a thin nail driven into the wall at an angle. It works in drywall and leaves only a small hole, so it is popular in rentals. Load ratings differ by product, so choose one for the frame's weight. |
| Drywall | The board used for most interior walls and ceilings. Plain screws do not hold well; for heavy frames, find a stud (the wood framing inside the wall) or use drywall anchors or hooks. |
| Even spacing | A layout where the side margins and the gaps between frames are all the same within the wall width. There are \(n + 1\) spaces, and the leftover width is divided equally among them. |
| Centering | A layout where you set the gap first and place the whole group in the middle of the wall. The side margins are equal, but they are a different value from the gap. |
| Pythagorean theorem | In a right triangle, base² + height² = hypotenuse². It is used here with half the wire as the hypotenuse and half the D-ring spacing as the base to find the rise of the peak \(r\). |
| Level | A tool that checks whether something is horizontal. Hold it on the top of the frame after hanging and straighten it. A phone level app works too. |
| Outside dimensions | The size of the frame including the molding. Enter the frame height and width in this calculator as outside dimensions, not the size of the picture or mat opening. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the quickest way forward.
| Units of length and measuring (Grades 2–4) |
|
| Addition, subtraction and halves (Grades 2–3) |
|
| Multiplication and equal sharing (Grades 3–4) |
|
| Expressions with variables (Grade 6) |
|
| Positive and negative numbers (Grades 6–7) |
|
| The Pythagorean theorem and square roots (Grade 8) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the top and bottom of the frame and the nail height
| Center height c (in) | 57 |
| Frame height H (in) | 24 |
| Top to hanging point w (in) | 2 |
| Top of frame T (in) | =B1+B2/2 |
| Bottom of frame B (in) | =B1-B2/2 |
| Nail height h (in) | =B4-B3 |
Table to find the height of the wire peak
| Wire length ℓ (in) | 20 |
| D-ring spacing s (in) | 16 |
| Top to D-rings d (in) | 8 |
| Rise of the peak r (in) | =SQRT((B1/2)^2-(B2/2)^2) |
| Top to hanging point w (in) | =B3-B4 |
Table to find the total width and side margins of a row (set gap)
| Wall width W (in) | 120 |
| Number of frames n | 3 |
| Frame width a (in) | 16 |
| Gap g (in) | 2 |
| Total width Tw (in) | =B2*B3+(B2-1)*B4 |
| Side margin m (in) | =(B1-B5)/2 |
| Center of frame 1 x1 (in) | =B6+B3/2 |
| Center of frame 2 x2 (in) | =B7+(B3+B4) |
| Center of frame 3 x3 (in) | =B8+(B3+B4) |
Table to find the gap for even spacing
| Wall width W (in) | 120 |
| Number of frames n | 3 |
| Frame width a (in) | 16 |
| Gap g (= side margin m) (in) | =(B1-B2*B3)/(B2+1) |
| Center of frame 1 x1 (in) | =B4+B3/2 |
After pasting, the upper rows of column B are the inputs and the lower rows are calculated automatically.
The first table (center 57 in, height 24 in, 2 in to the hanger) gives B4 = 69, B5 = 45 and B6 = 67 (the nail height). The second (20 in wire, 16 in spacing, 8 in to the D-rings) gives B4 = 6 and B5 = 2. The third (120 in wall, 3 frames, 16 in wide, 2 in gaps) gives B5 = 52, B6 = 34 and B7 to B9 = 42, 60, 78. The fourth, even spacing on the same wall, gives B4 = 18 and B5 = 26.
SQRT is the square root and "^2" is squared. For 4 or more frames, add x rows below the third table in the form "row above + (B3+B4)".
The first table (center 57 in, height 24 in, 2 in to the hanger) gives B4 = 69, B5 = 45 and B6 = 67 (the nail height). The second (20 in wire, 16 in spacing, 8 in to the D-rings) gives B4 = 6 and B5 = 2. The third (120 in wall, 3 frames, 16 in wide, 2 in gaps) gives B5 = 52, B6 = 34 and B7 to B9 = 42, 60, 78. The fourth, even spacing on the same wall, gives B4 = 18 and B5 = 26.
SQRT is the square root and "^2" is squared. For 4 or more frames, add x rows below the third table in the form "row above + (B3+B4)".
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the top and bottom of the frame and the nail height
| Center height c (in) | 57 |
| Frame height H (in) | 24 |
| Top to hanging point w (in) | 2 |
| Top of frame T (in) | =B1+B2/2 |
| Bottom of frame B (in) | =B1-B2/2 |
| Nail height h (in) | =B4-B3 |
Table to find the height of the wire peak
| Wire length ℓ (in) | 20 |
| D-ring spacing s (in) | 16 |
| Top to D-rings d (in) | 8 |
| Rise of the peak r (in) | =SQRT((B1/2)^2-(B2/2)^2) |
| Top to hanging point w (in) | =B3-B4 |
Table to find the total width and side margins of a row (set gap)
| Wall width W (in) | 120 |
| Number of frames n | 3 |
| Frame width a (in) | 16 |
| Gap g (in) | 2 |
| Total width Tw (in) | =B2*B3+(B2-1)*B4 |
| Side margin m (in) | =(B1-B5)/2 |
| Center of frame 1 x1 (in) | =B6+B3/2 |
| Center of frame 2 x2 (in) | =B7+(B3+B4) |
| Center of frame 3 x3 (in) | =B8+(B3+B4) |
Table to find the gap for even spacing
| Wall width W (in) | 120 |
| Number of frames n | 3 |
| Frame width a (in) | 16 |
| Gap g (= side margin m) (in) | =(B1-B2*B3)/(B2+1) |
| Center of frame 1 x1 (in) | =B4+B3/2 |
The same formulas as Excel (SQRT has the same name) work as is. Copy the whole table, paste it into cell A1 and change column B to your own numbers.
How to calculate it in Python
import math
# ---- One frame: nail height ----
center_height = 57 # target center height c (from the floor, in). Eye level
frame_height = 24 # frame height H (outside, in)
hanger_offset = 2 # top of frame to hanging point w (in)
top = center_height + frame_height / 2
bottom = center_height - frame_height / 2
hook_height = top - hanger_offset
print(f"Top: {top} in, bottom: {bottom} in, nail height: {hook_height} in")
# ---- Height of the wire peak (Pythagorean theorem) ----
wire_length = 20 # wire length l (in)
hanger_span = 16 # D-ring spacing s (in)
hanger_depth = 8 # top of frame to D-rings d (in)
rise = math.sqrt((wire_length / 2) ** 2 - (hanger_span / 2) ** 2)
offset_from_top = hanger_depth - rise
print(f"Rise of the peak: {rise:.1f} in, top to hanging point: {offset_from_top:.1f} in")
# ---- Row: set the gap and center on the wall ----
wall_width = 120 # wall width W (in)
count = 3 # number of frames n
frame_width = 16 # frame width a (in)
gap = 2 # gap g (in)
total_width = count * frame_width + (count - 1) * gap
margin = (wall_width - total_width) / 2
centers = [margin + frame_width / 2 + i * (frame_width + gap) for i in range(count)]
print(f"Total width: {total_width} in, side margin: {margin} in, frame centers: {centers}")
# ---- Row: space evenly across the wall (side margins = gaps) ----
gap_even = (wall_width - count * frame_width) / (count + 1)
centers_even = [gap_even + frame_width / 2 + i * (frame_width + gap_even) for i in range(count)]
print(f"Even gap: {gap_even} in, frame centers: {centers_even}")
It runs with the standard library only. math.sqrt() is the square root. Change the center height, frame size, hanger position and wall width at the top to your own numbers and run it. The list centers holds the distance from the left end of the wall to each frame center (in).
How to write it in LaTeX and other math languages (copy and paste)
Top and bottom of the frame and the nail height (one frame)
T = c + H ÷ 2, B = c − H ÷ 2, h = T − w
T = c + \frac{H}{2},\quad B = c - \frac{H}{2},\quad h = T - w
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>T</mi><mo>=</mo><mi>c</mi><mo>+</mo><mfrac><mi>H</mi><mn>2</mn></mfrac>
<mo>,</mo>
<mi>B</mi><mo>=</mo><mi>c</mi><mo>−</mo><mfrac><mi>H</mi><mn>2</mn></mfrac>
<mo>,</mo>
<mi>h</mi><mo>=</mo><mi>T</mi><mo>−</mo><mi>w</mi>
</mrow>
</math>
T = c + H/2, B = c - H/2, h = T - w
{c + H/2, c - H/2, c + H/2 - w}
T := c + H/2; B := c - H/2; h := T - w;
T = c + H/2; B = c - H/2; h = T - w; % c is the center height, H the frame height, w the top-to-hanging-point distance (in)
T = c + H/2, B = c − H/2, h = T − w
Height of the wire peak when pulled tight (allowing for slack)
r = √((ℓ ÷ 2)² − (s ÷ 2)²), w = d − r
r = \sqrt{\left(\frac{\ell}{2}\right)^{2} - \left(\frac{s}{2}\right)^{2}},\quad w = d - r
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>r</mi><mo>=</mo>
<msqrt>
<msup><mrow><mo>(</mo><mfrac><mi>ℓ</mi><mn>2</mn></mfrac><mo>)</mo></mrow><mn>2</mn></msup>
<mo>−</mo>
<msup><mrow><mo>(</mo><mfrac><mi>s</mi><mn>2</mn></mfrac><mo>)</mo></mrow><mn>2</mn></msup>
</msqrt>
<mo>,</mo>
<mi>w</mi><mo>=</mo><mi>d</mi><mo>−</mo><mi>r</mi>
</mrow>
</math>
r = sqrt((l/2)^2 - (s/2)^2), w = d - r
r = Sqrt[(l/2)^2 - (s/2)^2]; w = d - r
r := sqrt((l/2)^2 - (s/2)^2); w := d - r;
r = sqrt((l/2)^2 - (s/2)^2); w = d - r; % l is the wire length, s the D-ring spacing, d the top-to-D-ring distance (in)
r = √((ℓ/2)^2 − (s/2)^2), w = d − r
Total width and side margins of a row (set the gap and center)
Tw = n × a + (n − 1) × g, m = (W − Tw) ÷ 2
T_{w} = na + (n-1)g,\quad m = \frac{W - T_{w}}{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>T</mi><mi>w</mi></msub><mo>=</mo><mi>n</mi><mo>⁢</mo><mi>a</mi><mo>+</mo>
<mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>⁢</mo><mi>g</mi>
<mo>,</mo>
<mi>m</mi><mo>=</mo><mfrac><mrow><mi>W</mi><mo>−</mo><msub><mi>T</mi><mi>w</mi></msub></mrow><mn>2</mn></mfrac>
</mrow>
</math>
T_w = n a + (n - 1) g, m = (W - T_w) / 2
Tw = n*a + (n - 1)*g; m = (W - Tw)/2
Tw := n*a + (n - 1)*g; m := (W - Tw)/2;
Tw = n*a + (n - 1)*g; m = (W - Tw)/2; % n is the number of frames, a the frame width, g the gap, W the wall width (in)
T_w = na + (n − 1)g, m = (W − T_w)/2
Gap for even spacing across the wall (side margins = gaps)
g = (W − n × a) ÷ (n + 1), m = g
g = \frac{W - na}{n + 1},\quad m = g
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>g</mi><mo>=</mo>
<mfrac><mrow><mi>W</mi><mo>−</mo><mi>n</mi><mo>⁢</mo><mi>a</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac>
<mo>,</mo>
<mi>m</mi><mo>=</mo><mi>g</mi>
</mrow>
</math>
g = (W - n a) / (n + 1), m = g
g = (W - n*a)/(n + 1); m = g
g := (W - n*a)/(n + 1); m := g;
g = (W - n*a)/(n + 1); m = g; % W is the wall width, n the number of frames, a the frame width (in)
g = (W − na)/(n + 1), m = g
The \(x\) position of each frame center (nail)
xᵢ = m + a ÷ 2 + (i − 1) × (a + g)
x_{i} = m + \frac{a}{2} + (i - 1)(a + g)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mi>m</mi><mo>+</mo><mfrac><mi>a</mi><mn>2</mn></mfrac><mo>+</mo>
<mo>(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>⁢</mo>
<mo>(</mo><mi>a</mi><mo>+</mo><mi>g</mi><mo>)</mo>
</mrow>
</math>
x_i = m + a/2 + (i - 1)(a + g)
x[i_] := m + a/2 + (i - 1)*(a + g)
x := i -> m + a/2 + (i - 1)*(a + g);
k = 1:n; x = m + a/2 + (k - 1)*(a + g); % k is the frame number, m the margin, a the frame width, g the gap, n the number of frames
x_i = m + a/2 + (i − 1)(a + g)
How to have ChatGPT do the calculation
You are an assistant for working out where to hang pictures. Do the calculations below by actually running Python code, and base your answer only on the numbers from the output (do not calculate in your head or guess). I am hanging three frames, each 24 in tall and 16 in wide, in a row on a 120 in wide wall. The frame centers go 57 in above the floor. Each frame has a sawtooth hanger 2 in below the top. Find each of the following (in inches). 1. The heights of the top and bottom of the frames (top = center + height ÷ 2, bottom = center − height ÷ 2) and the nail height (top − 2) 2. With 2 in gaps and the group centered: the total width (3 × 16 + 2 × 2), the side margins ((120 − total width) ÷ 2) and the x position of each frame center (from the left end of the wall: margin + 8, then every 18) 3. With even spacing, where the side margins and gaps are all the same: the gap ((120 − 3 × 16) ÷ 4) and the x position of each frame center 4. If instead a 20 in wire is strung between D-rings 8 in below the top and 16 in apart, hung on one nail: the rise of the peak above the D-rings (√((20÷2)² − (16÷2)²)), the distance from the top to the peak, and the nail height then Show the formulas you used and the numbers from the output.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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