# How To Measure The Height Of A Building With A Barometer

## 1. Measure the barometric pressure at the top and at the bottom

Measure the barometric pressure (in mm of mercury) at the base of the building. Take the barometer to the roof of the building, and measure the pressure again. The difference tells you the weight of a column of air the same height as the building. Multiply the pressure difference (in mm of mercury) by the relative density of mercury compared to the air around the building. This gives you the height of the building (in mm). Use standard tables to convert to the required unit of measurement.

## 2. Drop the barometer and time how long it takes to fall

Take the barometer to the roof of the building. Walk to the edge. Drop the barometer over the edge and time how long it takes to hit the ground. You may either watch the barometer fall, or listen for it hitting the ground, depending on the height of the building and the accuracy required. Don't forget to correct for the speed of sound if listening for the crunch. Use the fact that height is gravity times the square of the time, divided by two to calculate height from the (known) gravity and the (measured) time.

## 3. Use the barometer as a measuring stick

Place the barometer upright against the wall. Mark the top of the barometer. Label the mark '1'. Move the barometer vertically so that the bottom of the barometer is at the mark. Mark the top of the barometer, labelling this mark '2'. Continue like this until you reach the top of the building. Multiply the number on the last mark by the height of the barometer. This will be accurate to within one barometer height. For greater accuracy, add one-half the height of the barometer to account for the portion above the last mark.

Note: For tall buildings, some extra equipment may be required to assist in climbing the wall.

## 4. Offer the barometer to the superintendent

Find the superintendent of the building. Offer him a deluxe display barometer if he will tell you the height of the building. If the superintendent is not available, or doesn't like barometers, try other parties such as the local survey officer, the original architect, or a member of the construction crew.

## 5. Measure the shadow of the barometer and the building

On a sunny day, measure the length of the shadow cast by the barometer, and the length of the shadow cast by the building. Multiply this ratio by the height of the barometer to get the height of the building.

## 6. Measure the shadow of the building, calibrated by the barometer

On a sunny day, place the barometer on the ground. Mark both ends. Stand the barometer upright on the mark closer to the sun, so the shadow will be approaching the other mark. Note the exact time when the shadow reaches the other mark.

On the following sunny day, mark the end of the shadow cast by the building at exactly that time. Measure the distance from there to the building. This is the height of the building.

Note: The accuracy of this technique depends on the number of days between consecutive sunny days.

## 7. Find a barometer with heights of local buildings on it

Go to all the local gift shops. Look for a fancy souvenir barometer, the kind which shows important local landmarks. Find one which shows the heights of local buildings and considers this building important enough to be listed. Use this barometer.

## 8. Compare the barometer height to the building height

Hold the barometer one foot in front of yourself and find a position where the building appears to be the same size as the barometer. Now measure the distance to the building (in feet) and multiply by the height of the barometer.

## 9. Trade the barometer for a long measuring tape

Go to a local shop and trade the barometer for the longest measuring tape they have. Take the tape onto the roof of the building. Holding one end, drop the other end over the edge of the building. Raise the measuring tape until the far end is just touching, not resting on, the ground. Read the height of the building from the measuring tape.

Note: For particularly tall buildings, this may require a particularly good hardware store.

## 10. Drop the barometer on the roof and on the ground

Hold the barometer straight in front of you and drop it. Measure, very carefully, how long it takes to hit the ground. Go up on the roof and hold the barometer in the same position. Drop it and measure, again very carefully, how long it takes to hit the roof. Since gravity falls off as the square of the distance from the centre of the planet, you can use the difference in times to calculate the height of the building relative to the distance from the base of the building to the centre of the planet. The local library can provide you with the distance to the centre of the planet in the required units.

Note: The ratio of the times is the same as the ratio of the distances from the drop points to the centre of the planet.

## 11. Use the barometer as a pendulum

Buy a rope long enough to reach the top of the building. Tie the barometer to one end and go up on the roof. Lower the barometer until it is exactly one inch above the ground. Holding the top of the rope at the top of the building, swing the barometer, and measure the period of the pendulum. From this you can calculate the length of the rope. Add one inch and you have the height of the building.

## 12. Drop the barometer on a windless day

Drop (and shatter) the mercury barometer at the base of the buidling on a windless day. Measure the increase in the mercury vapour concentration at the top of the building. Solve the diffusion equation to determine the distance from the shattered barometer to the top of the building.

## 13. Seal the building and fill with water

Place the barometer on the ground floor of the building. Seal all the building's doors and windows. Fill the building with water. Read the pressure measurement from the barometer. This gives the weight of a column of water the same height as the building. Use this and the ratio of the density of mercury to the density of water to calculate the height of the building.

Note: It is common courtesy to evacuate the building before using this technique.

## 14. Take the barometer to an airless world

If you have access to an airless world, take the building there. Throw the barometer horizontally off the building. If the barometer hits the ground, retrieve it and try again, throwing harder. The objective is to throw it hard enough to achieve a circular orbit. Once the barometer is in orbit around the planet, you can measure the period of the orbit. Compare this with the period of the orbit when you throw the barometer from the base of the building. Use this ratio and Kepler's laws to determine the height of the building (relative to the radius of the planet).

## 15. Clap the barometer and listen for the echo

(This one was contributed by one of our readers in October 2001 - thanks, Chandra!) Clap the barometer against the top of the building. Measure the time taken to hear the echo from the ground. Find the height of the building by multiplying half the echo delay by the velocity of sound.

### Where did the barometer question and answers come from?

This list of answers was compiled originally by a group of university students and extended by answers from the net. The question has been around for a very long time. You may want to read the original story.

Beauty is the first test; there is no permanent place in the world for ugly mathematics
Godfrey Harold Hardy, 1940

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Innumeracy, or sleight of number.
Some oddities of our language.
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