 Welcome to this metrology module area of circles and volume of cylinders. Metrology is the science of measurement. It plays a vital role in modern manufacturing. Every day, thousands of parts are produced in many industries, like aerospace and automotive. Parts must be produced to precise specifications. If those specifications aren't met, component parts won't fit together or work correctly. This can cost millions of dollars in product waste. Metrology is the key to maintaining mass production and creating a quality product. While this term may be unfamiliar to many people, it touches all of our lives. When you see your doctor, you want a precise measurement of the cholesterol level in your blood. When you're driving down the highway, you need your speedometer to accurately measure how fast you're going. When you're cooking dinner, you need your measuring cups to be accurate, so your recipe turns out great. The list of ways metrology impacts your life is endless, which is why understanding the science behind it is so important. Every metrologist uses many tools to ensure precision. In this module, we'll explore some commonly used formulas for area and volume. We'll find our areas and volumes by using formulas written in terms of the diameter of a circle, as opposed to the radius. We do this because common metrology tools can't directly measure the radius. Also, drawings typically dimension the diameter and not the radii of circular, cylindrical, and conical features. For those reasons and because of some tolerancing issues, metrologists often work with diameters when doing calculations. Let's start with reviewing how we calculate the area of a circle because we have to know this in order to calculate volume. The most common formula is π times the radius squared. In our math classes, we learn that the radius of a circle is half of its diameter. Knowing this means we can easily switch this formula, so it uses the circle's diameter instead of the radius. We replace r with the diameter divided by 2. We then square d over 2 and multiply that result by π. This leaves us with the area being equal to π times the diameter squared, all divided by 4. Now let's practice. Find the area of this circle that has a diameter of 7.5 feet. Use the π key on your calculator and round your final number to the hundredths. Press pause to work through this yourself and then press play to see the answer. We started by writing the formula for finding the area of the circle using its diameter. Then we replaced d with 7.5. To execute this calculation, we use the π key on our calculator, multiply that times 7.5, squared, then divide by 4. Then we rounded our final answer to the hundredths and attached the appropriate units, which in this case is square feet. Let's look at a more advanced example of finding the area of a circle that's also one you might encounter in your work. This is a bushing and we need to calculate the cross-sectional area. The bushing has an outside diameter of 40 mm and an inner diameter of 30 mm, and the cross-section looks like a ring. To calculate the area, we need to subtract the area of the small circle from that of the large circle. Watch as we perform this calculation. We evaluate the formula using the π button on our calculator to get the answer and we round that to the hundredths place. Finally, we add the correct unit of measurement, which in this case is square millimeters. Now that we know how to find the area of circles, we can use that information to find the volume of cylinders. A cylinder is a three-dimensional shape, consisting of two parallel circular bases joined by a curved surface. The center of the circular bases overlaps each other to form a right cylinder, which is used in many engineering applications, including the production of steel shafts, bushings, and piston engines. Cylinders are also used for many consumer products like soda cans, batteries, and wine corks. The most common formula for the volume of a cylinder is V equals πR squared times H. You can see that calculating the volume requires using not only the diameter, but also the height of the cylinder. We already know that the radius is half the diameter of any circle. If we replace R in the formula with the diameter divided by 2, we create a formula that finds the volume of a cylinder in terms of its diameter. This is what we use. Volume equals π times the diameter squared times the height and all divided by 4. In this example, we're only finding the approximate volume of the soda can since the top and bottom of the can are tapered. If we work this problem, the volume is approximately 25.48 cubic inches. Now let's see how this applies to manufacturing. Knowing the formula for the volume of a cylinder helps when you need to design a cylindrical object. The dimensions usually require a very high precision to ensure the part functions properly. In order to eliminate waste, the machine is often used scrap bar stock to calculate volume measurements. Let's work through the dimension measurements and volume calculations for this piece of scrap bar stock. In this image, you can see that the height is 6 1⁄8 inch. We use a micrometer to measure the diameter. In this case, we're using a 2 to 3 inch micrometer to find out the diameter is 2.5174 inches. Now, follow along as we make our volume calculation. Once we finish rounding, we find that the volume for this piece of scrap bar stock is approximately 30.486 cubic inches. Now, let's look at a more advanced problem. It's also one you may encounter in the real world. We need to find the weight of this steel shaft, but we first need to know its volume. The formula for finding weight is volume times density. To find the volume, we add together the volume of the two individual cylinders. Remember, the formula for the volume of a cylinder is V equals pi D squared times the height and all divided by 4. We use this formula twice by inserting the dimensions of each cylinder into the equations. Then, we multiply it by steel's density, which is 0.26 pounds per cubic inch. Finally, we round to the nearest tenth and find out that the weight of this steel shaft is 2.3 pounds. Finally, let's wrap up by looking at a few other formulas where diameter is often used. These include finding the circumference of a circle, total surface area of a cylinder, total surface area of a sphere, and volume of a sphere. We won't go into these formulas in detail, but it's good to be aware of these for future projects. In conclusion, metrology plays a vital role in modern manufacturing. Making sure your component parts are accurate and sure they fit and will work as components in bigger products. In this module, we explored how to find the area of circles and volume of cylinders. You've completed this metrology module, area of circles and volume of cylinders.