Today, I bring you an excerpt from my book, Math for Grownups.  Enjoy!  (Now you have your weekend project planned for you.)

Hanging pictures can be a tricky business. If you’re not careful, your foyer can look like a hall of mirrors—with crooked photos of your wedding party alongside drawings that your kid made in kindergarten. Not to mention the holes in the drywall from when you realized that you hung your college diploma so high up the wall that only a giant could read it.

Not exactly the look you were going for?

You may not want to face it, but a tape measure, pencil, and yes, even a level, are your best buddies in home decorating. And hanging anything on your walls is no exception. Let’s look at this in a bit more detail.

Mimsy Mimsiton is thrilled to have finally received the oil portrait of her dear Mr. Cuddles, a teacup poodle who is set to inherit her large fortune. The painting will look fabulous above the marble fireplace in the west-wing lounge of her mansion.

But drat! The museum curator Mimsy has on retainer is in Paris, looking for additions to Mimsy’s collection of French landscapes. (She’s redoing the upstairs powder room and wants just the right Monet to round out the décor.)

But the painting must be hung before Mr. Cuddles’s birthday party. His little poodle friends would be so disappointed not to see it! There’s no way around it; Mimsy’s poor, overworked House Manager must hang the painting herself.

Luckily, House Manager is no stranger to the DIY trend, and Butler will be there to help. The two meet in the lounge, where the painting has already been delivered—along with a stepladder, a tape measure, and a pencil. Once House Manager marks the spot, Handy Man will come along to safely secure the painting to the wall.

House Manager and Butler get to work. First they measure the painting: With the gilded frame, it’s 54″ tall and 60″ wide.

Next, they turn their attention to the space above the mantle. House Manager climbs atop the ladder, while Butler holds it steady. From the ceiling to the top of the mantle is 84″.  The width of the mantle is 75″.

Climbing down from the ladder, House Manager notes that the painting will certainly fit in the space allotted. She knows from experience that it is to be centered over the mantle. However, Mimsy will have a fit if the painting is centered vertically—between the ceiling and the mantle. No, the bottom of the painting must be exactly 12″ above the mantle.

So how high should Handy Man install the picture hanger?

To find out, House Manager must add 12″ to 54″ (the height of the painting). The top of the painting should be 66″ above the mantle.

House Manager grabs her tape measure again and removes the freshly sharpened pencil from behind her ear. Then she climbs the ladder. Starting at one end of the mantle, she measures 37½”—which is half the width of the mantle. She makes a barely visible pencil mark at that point.

Then from there, she measures up the wall to 64″. Again, she carefully makes a faint pencil mark.

If House Manager stopped here—leaving that small mark for Handy Man to hang the portrait—she’d probably be out of a job. That’s because she’s merely marked the top of the frame, not where the hanger should be secured.

She descends the ladder and goes back to the portrait. Turning it around, she notices the picture wire that has been stretched from one side to the other. She hooks her finger under the center of the wire and pulls up gently—creating an angle, as if the picture wire were hanging on a nail. Now an angle is a two-dimensional figure formed by two lines (called rays) that share a common point. Hereʼs an easier way to remember this: An angle looks like a V.

If she can measure the distance from the top of the frame to the vertex—the point where two sides of an angle meet—she’ll be in business.

There’s just one more thing to consider: Is the vertex of the angle too far to the left or too far to the right?  For the painting to hang straight and be centered on the mantle, the vertex must be located at exactly half the width of the portrait.

House Manager uses her tape measure to find the length of each leg of the angle. In other words, she measures the distance from one end of the picture wire to the vertex of the angle and then the distance from the vertex of the angle to the other end of the wire. If the vertex is centered properly, the legs of the angle will have the same length.

Moving her finger ever so slightly, House Manager centers the vertex of the wire angle—and measures from that point to the top of the picture frame: 9″.

She now can make the final mark for Handy Man. She climbs the ladder for the third time and measures 9″ from the mark she made earlier. Again, being very careful, she makes a tiny mark on the wall.

House Manager’s work is done. If anything goes wrong now, it’s Handy Man’s fault.

She folds up the ladder and gathers her supplies. Then she’s off to order beef cupcakes for Mr. Cuddles’s party.

Any questions?  Ask them in the comments section.