Math Unit 9 Measurements

To find the area of irregular shapes

break the shape up into smaller common shapes.

Area of a circle is found using the formula:

A = πr2 (where r = radius)

Volume is a three-dimensional measurement.

As such, a change in all dimensions by a factor of n will have an effect of n3.

To do a metric to metric conversion, you should be familiar with the multiples of 10 that correspond to each prefix (ex: 1 kilometer = 1000 meters)

For all other types of conversions, the conversion rate will most likely be given to you.

In math, the word tolerance is defined as the amount of error that is allowed.

Formally, accuracy is considered to be ±½ of the smallest measuring unit. Therefore, a ruler marked in ¼ inch increments, for example, is only accurate to one-eighth of an inch

high precision, low accuracy

Has repeated the darts in the same place, but is not hitting the bullseye

Calculating area, Method 1: When using the grid, area is simply the number of squares contained inside the figure. By counting, the areas of the figures are:

I: A = 9 units squared; II: A = 18 units squared; III: A = 14 units squared; IV: A=20 units squared. Method 2: Instead of counting individual boxes, one can multiply the length of two consecutive sides to find each rectangle's area.

If the dimensions were increased by a scale factor of 3, the perimeter would triple and the area would increase by a factor of 9.

In general, a figure can be increased proportionally by multiplying by a scale factor n, where n >1. Such a change will increase the perimeter by a factor of n and will increase the area by a scale factor of n2

A common example of this is the speed we drive, 60 miles per hour, or the gas our cars use, 13 miles per gallon. Unit rates always have 1 as a part of the ratio

In the example given, speed is the number of miles driven in one hour and gas usage is rated at miles driven in one hour.

Sometimes polygons have non standard shapes.

In this case a formula alone will not work and the polygon must be divided into multiple simpler shapes and then added together to find the area.

Surface area is the area on the outside of a three-dimensional object. It can be measured as lateral, which accounts for only the sides of the object.

It can also be measured as total, which is the lateral area plus any top or bottom of the object.

Notice that the area of Figure V has the same side lengths as Figure IV.

The area of a triangle is ½ the area of a related rectangle.

The lateral surface area of a solid figure is the sum of the areas of all faces excluding the base(s)

The lateral surface area, or lateral area, of a can such as a soup can would be that part of the can covered by the label. By contrast, the surface area, or total surface area, of the can would be the entire metal surface of the can.

Mass is constant.

The mass of an object will be identical on Earth, on the moon, in space, or even underwater, as the mass of an object relates to how much matter it contains.

Not every unit will make sense to use in every type of measurement situation. Consider 3 different units that could be used to measure an adult person's weight: 2880 ounces 180 lbs .09 tons

The measurement in ounces is a very large number, and the measurement in tons is a very small number, so the measurement is lbs in the easiest to work with.

In this context, use of b (lowercase b) stands for the base in a two-dimensional figure.

The measurement of b is a one-dimensional measurement, also known as length (as in, the area of a rectangle can be found by length × width).

The English system (sometimes known as the Imperial system) is a system of measurement used in the United States using units such as feet, pounds and ounces.

The metric system (sometimes known as SI) is a system of measurement used throughout the world in which the basic units are the meter, the gram, and the liter

Nets can also be used to compute the surface area of a three dimensional figure.

The problem tells us that this net makes a cube. A cube is a three dimensional shape with equal length sides.

Geometry problems often ask to solve for missing angles or to determine information about triangles based on information given about angles.

The sum of the interior angles of a triangle is always 180°

There is no standard formula to determine the area of this shape. Decomposing the figure into two common shapes makes the math easy.

There's a bottom rectangle of dimensions 6 cm by 4 cm. On top there is a triangle with base of 6 cm and height of 2 cm.

a scale that measures weight in grams is accurate to ±1/2 gram; when using a protractor marked in degrees, the accuracy is ±½ degree

Therefore, the smaller the mark on a measuring tool, the greater the precision.

The volume of three-dimensional figures is the measure of the total three dimensional space contained in an object.

These objects include spheres, rectangular and triangular prisms, rectangular and triangular pyramids, cylinders, and cones.

A proportion is two equal fractions.

This method sets up a proportion using the conversion rate as one of the fractions, and the given information and variable as the other fraction

Protractors are used to measure angles.

To use a protractor, align one line horizontally with the 0 degree mark. Then read the measure of the other line. When measuring angles using a protractor it is important to consider whether the angle is acute or obtuse

Scales are used to measure weight

To use a scale, turn the scale on, make sure the display reads 0 (tare), and then place the object you wish to measure on the scale.

In general, volume formulas for solids are some form of the expression: Base × height

Use of the uppercase B refers to the area of the base in a three-dimensional figure and, because it is an area itself, is a two-dimensional unit

Since one edge is labelled as 2cm, the surface area of each face must be 2\times×2 = 4cm squared. There are 6 total faces. Therefore the total surface area is 6 \times× 4cm squared = 24 cm squared

When calculating surface area of 3D figures, sketch the net and find the area of each face.

Precision is

a way of determining/discussing what level of accuracy is possible based on the tools used.

Weight is affected by gravity

and will change based on gravitational force. Therefore, the weight of an object on Earth is considerably heavier than the same object's weight on the moon or in space.

Rulers

are used to measure the length of objects. To use a ruler, line the end that says zero up to one end of the object. Length can be measured in inches, centimeters, or millimeters.

To determine the perimeter of each rectangle:

begin at one vertex, count the lengths of each of the sides, and add the lengths together. Using this approach, the perimeters of these figures are found to be:

The circumference of a circle can be found by the formula:

can be found by the formula: Circumference of a circle (c) = 2πr = πd

High precision, high accuracy

darts hit the center of the target

A scale factor of n <1

decreases the perimeter by a factor of n, decreases the area by a factor of n2, decreases the volume by a factor of n3

low precision, high accuracy

has all darts close to the target, but not hitting it in the same place

Since a key tenet of the scientific method is for the results to be tested and retested by other scientists, It is important for mathematicians and scientists to have a universally-communicable method of reporting data

his method is known as the International System of Units (SI) or the metric system.

Perimeter is a one-dimensional measurement, so it will always be labeled in units such as:

inches (in.) feet (ft.) miles (mi.) millimeters (mm) centimeters (cm)

A scale factor of n >1

increases the perimeter by a factor of n, increases the area by a factor of n2, increases the volume by a factor of n3

Low Precision, Low accuracy

not repeating, not in the same place

To solve using proportions

set two fractions equal to each other and cross multiply.

The area is the amount of surface inside of a figure. Area is a two-dimensional measurement (length × width, for example) and is measured in square units, such as:

square inches (in.2) square miles (mi.2) square centimeters (cm2) square kilometers (km2)

The metric system is based on factors of ten

the base units include meters, liters, and grams, and each other unit is given a prefix based on its factor of ten. To convert between units, all that is required is to multiply or divide by ten.

r = radius

the distance from the center of a circle to a point on the circle

d = diameter

the length of a segment that goes from one point on a circle through its center, and continues on to another point on the other side of the circle.

. If the dimensions of a figure are multiplied by a scale factor n where 0 < n < 1, then

the original figure will decrease proportionately by a scale factor of n with respect to perimeter and by n2 with respect to area.

In general, when the figure's dimensions are increased by a scale factor of 2

the perimeter also increases by a factor of 2, but the area increases by a factor of 22 or 4.

Unit rates are similar to ratios in that they show

the relationship between two values

A measurement is a number that shows

the size or amount of something. Because of the limitations of human (and machine) error in using measurement tools, measurement is never exact; it is always an approximation.

Margin of error increases when multiple measurements are taken and combined. One illustration of this idea is that, using a foot-long ruler measured with increments of 1/16 in, a student determines that the length of the room is 15 feet and its width is 12 feet

therefore, the perimeter of the room should be 54 feet [perimeter is the total distance around the room, so 2(15) + 2(12) = 30 + 24 = 54]. However, with the smallest unit of measure on each ruler being 1/16 in, each use of the ruler brings in a margin of error of 1/32 in

Basic types of measurement include:

time, temperature, money, speed, acceleration, length, area, weight, mass, capacity (volume)

The surface area of a solid figure is found by determining the area of each face of the figure, then adding these areas together to get a

total surface area.

Essential concepts that must be understood in the subject of measurement are:

types of measurement, available units for the English customary system and the metric system, which unit will be most appropriate for a particular task, conversion from one unit to another, the margin of error for a given measurement tool

While the unit fractions used to convert units within a measurement system can be relied upon completely

unit fractions for use in converting between measurement systems can be seen as slightly inaccurate.