5 - Geometry and Measurement Concepts

Question 1

Define:

measurement unit

Answer 1

A standard used to measure a physical quantity, like length or mass.

Measurement units are essential for ensuring that measurements can be easily understood and compared.

Question 2

Identify:

4 common measurement units for length.

Answer 2

1. Millimeters (mm)
2. Centimeters (cm)
3. Meters (m)
4. Kilometers (km)

These units vary in size and are used based on the dimension of the object being measured.

Question 3

Explain:

When should millimeters be used?

Answer 3

To measure very small objects.

Examples include the width of a pencil, the length of a grain of rice, or the width of a paperclip.

Question 4

Identify:

What measurement unit should be used for objects larger than 10 mm but smaller than a meter?

Answer 4

centimeters

(cm)

There are 10 millimeters in every centimeter.

Question 5

Identify:

What is the most appropriate measurement unit for larger objects such as rooms or buildings?

Answer 5

meters

(m)

Meters are used for measuring lengths and heights of everyday structures.

Question 6

Identify:

3 measurement units for mass.

Answer 6

1. Milligrams (mg)
2. Grams (g)
3. Kilograms (kg)

Different units are used based on the size of the mass being measured.

Question 7

Identify:

Which unit would you use to measure a single grain of rice in terms of mass?

Answer 7

milligrams

(mg)

The mass of a single grain of rice is about 25 mg.

Question 8

Identify:

What unit typically is used for measuring mass of food items?

Answer 8

grams

(g)

There are 1,000 milligrams in every gram, making it suitable for everyday food items.

Question 9

Define:

measurement

Answer 9

It quantifies the physical characteristics of an object, such as its weight or length.

Question 10

Explain:

How are measurements treated in mathematical operations?

Answer 10

Like variables when performing operations such as adding, subtracting, multiplying and dividing.

Question 11

Identify:

What step is required before adding or subtracting measurement units?

Answer 11

Check to make sure measurement units are the same.

Question 12

Explain:

If one table measures 3 feet and another measures 36 inches, how can you add them?

Answer 12

Convert 36 inches to feet, resulting in 3 feet, then add: 3 feet + 3 feet = 6 feet.

Question 13

Explain:

What happens to the measurement units when multiplying two measurements?

Answer 13

The result is in squared units.

Question 14

Identify:

3 components needed to convert units in the metric system.

Answer 14

1. What you’re starting with.
2. What you want to end up with.
3. The conversion factor.

This is analogous to making a smoothie: the ingredients, the smoothie itself, and the blender used to mix them.

Question 15

Identify:

The conversion factor for converting kilometers to meters.

Answer 15

1 kilometer = 1,000 meters

Question 16

Identify:

4 steps to set up a conversion problem in the metric system

Answer 16

1. Place the starting unit first
2. Add the conversion factor
3. Add the equals sign
4. Add the desired unit.

Example: (2,000 m / 1) * (1 km / 1,000 m) = km.

Question 17

Identify:

The conversion factor for converting milliliters to liters.

Answer 17

1 liter = 1,000 milliliters

Question 18

Explain:

What is the customary system of measurement?

Answer 18

The standard units of measurement for mass, length, capacity and temperature used primarily in the United States.

Also known as the U.S. customary system.

Question 19

Identify:

The countries that primarily use the customary system of measurement.

Answer 19

• United States
• Liberia
• Myanmar

Question 20

Identify:

The main difference between the metric system and the customary system.

Answer 20

The customary system is based on various units with seemingly random rates of conversion, while the metric system is based on the number 10 and powers of 10.

This makes the metric system easier to convert between units.

Question 21

Identify:

The primary units of length in the U.S. customary system.

Answer 21

• Inches (in)
• Feet (ft)
• Yards (yd)
• Miles (mi)

Examples include 12 inches in a foot and 3 feet in a yard.

Question 22

Identify:

The primary units of mass in the U.S. customary system.

Answer 22

• Ounces (oz)
• Pounds (lb)
• Tons (T)

For example, 16 ounces equal 1 pound and 2,000 pounds equal 1 ton.

Question 23

Identify:

The primary units of capacity in the U.S. customary system.

Answer 23

• Ounces (oz)
• Cups (c)
• Pints (pt)
• Quarts (qt)
• Gallons (gal)

For instance, 8 ounces equal 1 cup and 4 quarts equal 1 gallon.

Gallon Man is a fun way for students to see how different units of capacity are related.

Question 24

Identify:

The temperature scale used in the U.S. customary system.

Answer 24

Fahrenheit scale

Question 25

# Explain: **Rule** for **converting** from a **smaller unit to a larger unit** in the **customary system**

Answer 25

Divide by how many of smaller unit into larger unit. ## Footnote For example, to convert 60 inches to feet, divide by 12.

Question 26

# Explain: **Rule** for **converting** from a **larger unit to a smaller unit** in the customary system?

Answer 26

Multiply by larger unit. ## Footnote For instance, to convert 3 pints to cups, multiply by 2.

Question 27

# Identify: What year was the current **U.S. Customary System** was formally established?

Answer 27

1832

Question 28

# Identify: How many **feet** are in **one mile**?

Answer 28

5,280 feet

Question 29

# Identify: How many ounces are in 1 pound?

Answer 29

16 ## Footnote This is a standard conversion in the customary system.

Question 30

# Explain: What is the **equation** to convert 340 **ounces to pounds**?

Answer 30

340 ÷ 16 = 21 remainder 4. ## Footnote This illustrates the process of converting ounces to pounds in the customary system.

Question 31

# Identify: How many **gallons** are in **12** **quarts**?

Answer 31

3 gallons ## Footnote Since 4 quarts equal 1 gallon, you divide 12 by 4.

Question 32

# Identify: What is the conversion for **32 yards in feet**?

Answer 32

96 feet ## Footnote This is calculated by multiplying 32 yards by 3 feet per yard.

Question 33

# Explain: How many **inches are in 2.5 yards**?

Answer 33

90 inches ## Footnote The conversion involves first changing yards to feet and then feet to inches.

Question 34

# Identify: The **fundamental units** in measurement.

Answer 34

* Distance * Mass * Time

Question 35

# Define: blueprint

Answer 35

A **scaled drawing** of physical objects. ## Footnote Blueprints are often used in design and architecture.

Question 36

# Identify: The most commonly used **scale** for **blueprints**.

Answer 36

1/4 inch : 1 foot

Question 37

# Explain: What is the step needed to **convert a blueprint** dimension to feet?

Answer 37

Multiply the blueprint measurement by 4. ## Footnote This is based on the scale of 1/4 inch : 1 foot.

Question 38

# Explain: How long would it take to cover **120 miles at 130 km/hour**?

Answer 38

Approximately **1.5 hours** (1 hour and 30 minutes).

Question 39

# Identify: What is **unit** for measuring **speed**? | Context of driving a motor vehicle.

Answer 39

distance per time ## Footnote e.g., miles/hour

Question 40

# Explain: Convert **kilometers per hour to miles per hour** for travel estimation.

Answer 40

Use the conversion factor that **100 km/hour is close to 60 miles/hour**. ## Footnote This approximation aids in quick estimations.

Question 41

# Explain: Significance of **dimensional cancelling** in unit conversions.

Answer 41

Helps ensure the correct units are used in calculations. ## Footnote This process simplifies the conversion process.

Question 42

# Identify: The two basic **systems of measurement**.

Answer 42

1. International System (SI) 2. Imperial System ## Footnote SI is commonly known as the metric system.

Question 43

# Identify: The SI units for length.

Answer 43

* Millimeter (mm) * Centimeter (cm) * Meter (m) * Kilometer (km) ## Footnote These are also the most ommon measurement units for length.

Question 44

# Define: precision | Context of measurement.

Answer 44

The ability to repeat the measurement multiple times and get the same answer each time.

Question 45

# Explain: How to determine the **increment on a scale**.

Answer 45

Subtract the smaller number from the larger number between two consecutive markings.

Question 46

# Identify: The **units** used to measure **weight**.

Answer 46

* grams * kilograms * ounces * pounds

Question 47

# Identify: The freezing point of water in Celsius.

Answer 47

0 degrees C

Question 48

# Identify: The boiling point of water in Fahrenheit.

Answer 48

212 degrees F

Question 49

# Explain: Importance of **precision** in **medical instruments**.

Answer 49

To accurately measure a dose repeatedly.

Question 50

# Define: multimeter

Answer 50

**Instrument** used by electricians to test voltage, amps and resistance.

Question 51

# Identify: The **measurement units** a **multimeter** tests.

Answer 51

* voltage * amps * resistance

Question 52

# Identify: What is the increment division on a **multimeter**?

Answer 52

0.5 ## Footnote It is scaled with large numbers in increments of 10.

Question 53

# Explain: What is a **scale drawing**?

Answer 53

An accurately sized drawing that is either smaller or larger than real life. ## Footnote Scale drawings help to represent objects that are difficult to comprehend in their actual size.

Question 54

# Explain: What does the **legend** in a scale drawing indicate?

Answer 54

How much smaller or larger things in the drawing are. ## Footnote For example, it may state that 1 inch on the map equals 50 miles in real life.

Question 55

# Explain: How can you find the distance between two points on a map?

Answer 55

Measure the number of bars on the legend. ## Footnote Each bar on the legend represents a specific real-life distance, such as 50 miles.

Question 56

# Explain: The relationship between a **scale drawing** and **math**.

Answer 56

Math is used to determine how much smaller or larger to draw things. ## Footnote Accurate scaling requires consistent proportions in the drawing.

Question 57

# Explain: What does "***1 inch:12 inches***" signify in a model drawing?

Answer 57

Every inch in the drawing equals 12 inches in real life. ## Footnote This helps builders understand the actual size of the object.

Question 58

# Define: perimeter

Answer 58

The measure of the path or distance around the edge of two-dimensional geometric shapes, such as polygons and circles. ## Footnote Polygons are closed shapes with at least three straight sides and angles.

Question 59

# Explain: The difference between **perimeter** and **circumference**.

Answer 59

* Perimeter refers to **polygons**. * Circumference refers to **circles**. ## Footnote The distance around a circle is specifically called circumference.

Question 60

# Identify: The formula for finding the perimeter of a regular polygon.

Answer 60

Perimeter = n × a ## Footnote *n* is the number of sides. *a* is the length of one side. Regular polygons are equilateral and equiangular.

Question 61

# Define: irregular polygon

Answer 61

Any polygon that is not regular, meaning it is either not equilateral or not equiangular. ## Footnote Concave polygons are considered irregular due to unequal interior angles.

Question 62

# Identify: The universal formula for finding the **perimeter of any polygon**.

Answer 62

Perimeter = s1 + s2 + s3 + ⋯ + sN ## Footnote *si* is the length of the *i^th* side. This formula applies to both regular and irregular polygons.

Question 63

# Explain: What is the perimeter of an L-shaped irregular polygon with side lengths 12, 10, 8, 5, 3 and 2 units?

Answer 63

40 units ## Footnote Calculated as 12 + 10 + 8 + 5 + 3 + 2.

Question 64

# Define: equilateral triangle

Answer 64

A type of regular polygon with three congruent sides and angles. ## Footnote All sides and angles are equal in measure.

Question 65

# Identify: The formula for the perimeter of an equilateral triangle.

Answer 65

Perimeter = 3 × s ## Footnote *s* is the length of one side. Example: For a side length of 7 units, Perimeter = 3 × 7 = 21 units.

Question 66

# Identify: The perimeter of a square.

Answer 66

Perimeter = 4 × s ## Footnote *s* is the length of one side. Example: For a side length of 5 units, Perimeter = 4 × 5 = 20 units.

Question 67

# Define: rectangle

Answer 67

An equiangular quadrilateral composed of four right angles, with opposite sides parallel and congruent. ## Footnote A square is a specific type of rectangle.

Question 68

# Identify: The formula for the perimeter of a rectangle.

Answer 68

Perimeter = 2(l + w) ## Footnote *l* is the length. *w* is the width. Example: For length 12 units and width 15 units, Perimeter = 2(12 + 15) = 54 units.

Question 69

# Identify: 2 characteristics of regular polygons

Answer 69

1. equilateral 2. equiangular ## Footnote Examples include squares and equilateral triangles.

Question 70

# Define: quadrilateral

Answer 70

A four-sided polygon. ## Footnote Includes shapes like squares and rectangles.

Question 71

# Identify: The formula for finding the **circumference of a circle**.

Answer 71

Circumference = 2πr ## Footnote *r* is the radius of the circle. Example: For a radius of 8 units, Circumference ≈ 50.27 units.

Question 72

# Define: circumference

Answer 72

The **perimeter of a circle**, the curved length around it. ## Footnote It is equal in length to a straight line that matches the curved length.

Question 73

# Define: area of a circle

Answer 73

The measurement of the space inside the circle. ## Footnote It describes the amount of two-dimensional space that a circle occupies.

Question 74

# Identify: Formula to calculate the area of a circle.

Answer 74

A=πr² ## Footnote *π* is approximately 3.14. *r* is the radius of the circle.

Question 75

# Identify: Formula to calculate the **area of a circle using the diameter**.

Answer 75

Area=π(d/2)² ## Footnote The diameter must be divided by 2 to find the radius before applying the formula.

Question 76

# Identify: Formula for calculating the **circumference of a circle using the diameter**.

Answer 76

C=πd

Question 77

# Identify: Formula for calculating the **circumference of a circle using the radius**.

Answer 77

C=2πr ## Footnote This formula uses the radius to determine the circumference.

Question 78

# Explain: The relationship between the **radius** and **diameter** of a circle.

Answer 78

Diameter is twice the radius (**d=2r**). ## Footnote The radius is the distance from the center to the edge, while the diameter spans from one edge to the opposite edge through the center.

Question 79

# Define: **chord** in a circle

Answer 79

A line segment connecting any two points on the circle's edge. ## Footnote It runs through the circle but does not necessarily pass through the center.

Question 80

# Define: **tangent** in relation to a circle

Answer 80

Line that touches the circle at exactly one point. ## Footnote It does not intersect the circle at any other point.

Question 81

# Define: **arc** of a circle

Answer 81

A segment of the circumference between two points of a chord. | A portion of a circle's total circumference.

Question 82

# Define: **secant** in relation to a circle

Answer 82

A line that intersects the circle at two points. ## Footnote It extends beyond the endpoints of a chord.

Question 83

# Define: **segment** of a circle

Answer 83

A section of the area inside a circle split by a chord. ## Footnote It represents only a portion of the total area of the circle.

Question 84

# Define: semicircle

Answer 84

Half of a circle. ## Footnote It can be derived by dividing the area of a full circle by 2.

Question 85

# Identify: The common **units of length** used to measure the **radius** or **diameter of a circle**.

Answer 85

* Inches * Feet * Yards * Millimeters * Centimeters * Meters ## Footnote It is important to stay consistent with the unit system (imperial or metric) throughout measurements.

Question 86

# Identify: The formula to calculate the **area of a circle using its circumference**.

Answer 86

A=π(C/2π)² ## Footnote This formula is derived by solving the circumference formula for radius and substituting it into the area formula.

Question 87

# Identify: The value of **pi (π)**.

Answer 87

Approximately 3.14. ## Footnote It represents the ratio of the circumference of a circle to its diameter.

Question 88

# Identify: The **area of a square** with side length s.

Answer 88

s²

Question 89

# Explain: How do you calculate the **perimeter of a rectangle**?

Answer 89

2 * w + 2 * h

Question 90

# Identify: The formula for the **area of a triangle**.

Answer 90

1/2 * b * h

Question 91

# Identify: Formula for calculating the **area of a circle with radius r**.

Answer 91

π * r²

Question 92

# Identify: Formula for calculating **volume**.

Answer 92

length * width * height

Question 93

# Identify: **3D shapes** with the same cross-section from top to bottom.

Answer 93

* Cylinder * Cube * Prism

Question 94

# Identify: Volume formula for a **cylinder**.

Answer 94

h * (π * r²)

Question 95

# Identify: The volume formula for a **cube** with side length *s*.

Answer 95

s * s * s | Or s³.

Question 96

# Identify: Volume formula for a **sphere**.

Answer 96

4/3 * π * r³

Question 97

# Explain: Calculate the **volume of a hemisphere**.

Answer 97

1/2 * (4/3 * π * r³) or 2/3 * π * r³

Question 98

# Identify: What is the volume formula for a **cone**?

Answer 98

h/3 * (area of the base)

Question 99

# Identify: What is the volume formula for a **pyramid**?

Answer 99

h/3 * (area of the base) ## Footnote The volume formulas for pyramids and cones are the same.

Question 100

# Explain: Perimeter of a **semicircle**.

Answer 100

One-half the perimeter of a circle with the same radius. ## Footnote The perimeter of a circle with a radius of 5 would be expressed as: **2 * 5 * pi**.

Question 101

# Explain: What is the first step to find the **perimeter of a complicated shape**?

Answer 101

Break it down into smaller, manageable pieces. ## Footnote This approach simplifies the calculation and helps in accurately determining the perimeter.

Question 102

# Define: area | in geometry

Answer 102

The size of an object's surface; the space a flat object occupies. ## Footnote Example: A blue object taking up 9 blocks measures 9 cm².

Question 103

# Explain: What is the **formula** for the **area of a rectangle**?

Answer 103

Area = lw ## Footnote *l* is the length. *w* is the width.

Question 104

# Explain: What is the **formula** for the **area of a triangle**?

Answer 104

Area = 1/2 bh ## Footnote *b* is the base. *h* is the height.

Question 105

# Explain: What is the **formula** for the **area of a parallelogram**?

Answer 105

Area = bh ## Footnote *b* is the base. *h* is the height.

Question 106

# Explain:: What is the **formula** for the **area of a trapezium**?

Answer 106

Area = 1/2 (a + b)h ## Footnote *a* and *b* are the two bases. *h* is the height.

Question 107

# Define: volume

Answer 107

The amount of three-dimensional space that an object takes up, measured in cubic units. ## Footnote Common units include cubic meters, cubic feet, liters, and gallons.

Question 108

# Explain: How does **volume** relate to everyday items?

Answer 108

Many items are sold according to their volume, such as: * Gasoline (gallons) * Milk (gallons) * Soda (liters or ounces) * Soil (cubic yards) ## Footnote Volume is more relevant than mass or density in these contexts.

Question 109

# Identify: What is the formula for the volume of a cube?

Answer 109

V = l³ ## Footnote *l* = length. This formula is one of the simplest for calculating volume.

Question 110

# Identify: Formula for the **volume** of a **rectangular prism**.

Answer 110

V = l * w * h ## Footnote *l* = length. *w* = width. *h* = height. This is another basic volume formula.

Question 111

# Define: unit rate

Answer 111

Ratio where the denominator is 1 unit of the second quantity. ## Footnote Examples include speed of 45 miles per hour or a pay rate of $10 per hour.

Question 112

# Identify: Formula for calculating **speed as a unit rate**.

Answer 112

r = d/t ## Footnote Where d = distance traveled, t = amount of time traveled, and r = rate of speed.

Question 113

# Explain: How can you **remove fractions** when finding a **unit rate**?

Answer 113

Multiply both sides of the equation by *b*. ## Footnote This leads to the formula: **a = b × r**.

Question 114

# Identify: The first step in **solving unit rate problems**.

Answer 114

Set up a rate table. ## Footnote The table should include the rate formula and the situations given.

Question 115

# Identify: The formula to find the **rate of miles per gallon**.

Answer 115

r = d/g ## Footnote Where d is the distance driven and g is the gallons of gas used.

Question 116

# Identify: The formula used to determine the time taken for two trains traveling towards each other

Answer 116

d = r × t ## Footnote Where *d* is the distance, *r* is the speed, and *t* is the time.

Question 117

# Identify: Examples of **rates** found in daily life.

Answer 117

* Miles per hour. * Amount of money earned per time period. * Amount of precipitation per time period. ## Footnote These describe how one quantity changes in relation to another.

Question 118

# Identify: What is a key difference between a **rate** and a **simple distance measurement**?

Answer 118

* A rate compares two different changes. * Distance measures one quantity. ## Footnote For instance, the distance between two cities is not a rate.