In plain English: HSG.MG.A.2 is the Common Core geometry standard that asks students to apply density based on area and volume in modeling situations, such as persons per square mile or BTUs per cubic foot. Students find an area or volume with geometry, divide a quantity by it, or multiply a density by it to find a total, keeping the units straight. It is usually taught in high school Geometry.
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
Common Core State Standards for Mathematics · Domain: Modeling with Geometry (MG) · Cluster: Apply geometric concepts in modeling situations Also written as HSG-MG.A.2 or G-MG.2 · Official standard
Students apply the idea of density, an amount per unit of area or per unit of volume, in modeling situations. On the area side they work with population density, wildlife and planting densities, and seeding or cooling rates per square foot. On the volume side they work with mass density, the energy content of natural gas in BTUs per cubic foot, and the heat needed to warm the air in a room.
Geometry supplies the size: students model a region as a rectangle, triangle or circle, or an object as a prism, cylinder or sphere, and compute its area or volume before using the density. Every problem uses one relationship in three forms, and unit conversions for square and cubic units are practiced throughout.
Learning Objectives
By the end of this lesson, students will be able to:
Compute a density based on area, such as persons per square mile, from a total and a region modeled with a geometric shape
Compute a density based on volume, such as grams per cubic centimeter or BTUs per cubic foot, from a total and a volume
Use a known density to find a total amount or a needed size
Convert square and cubic units correctly within density problems
Decide whether a situation calls for an area-based or a volume-based density
Prior Knowledge Required
Students should already be comfortable with:
Unit rates and rate language 6.RP.A.2
Unit rates with fractions and different units 7.RP.A.1
Area and volume of rectangles, triangles, circles, prisms and cylinders 7.G.B.6
Volume formulas for cylinders, cones and spheres HSG.GMD.A.3
Using units to guide a multi-step problem HSN.Q.A.1
Show Diagram 1 without its bottom labels. Tell students each dot stands for 1,000 people.
Warm-Up Prompt
"Both towns have 24,000 people. Which town is more crowded? How could you put a number on 'crowded' so that a town of any size can be compared with any other?"
Students count 2 dots per square mile in Town A and 1 in Town B. Guide them to the idea of people per square mile: 24,000 ÷ 12 = 2,000 and 24,000 ÷ 24 = 1,000. Define density as an amount per unit of area or per unit of volume, and say that the whole lesson uses one relationship in three forms: density = amount ÷ size, amount = density × size, and size = amount ÷ density.
Direct Instruction20 minutes
Sort densities into two families and write both lists on the board:
Densities based on area and on volume
Based on
Examples
Units
Area
Population density, trees per acre, seed per lawn area, cooling per floor area
persons per mi², trees per acre, lb per 1,000 ft², BTU per hour per ft²
Volume
Mass density, energy in a fuel, heat needed by air
g per cm³, kg per m³, lb per ft³, BTU per ft³
Area: population density
An invented county has 612,000 residents and an area of 1,275 square miles. Find its population density.
Equation: 612,000 ÷ 1,275 = 480 persons per square mile
Area: working backward
A city is modeled as a rectangle 6 mi by 4.5 mi, with a density of 3,200 persons per square mile. Estimate its population.
Equation: Area = 27 mi²; population = 3,200 × 27 = 86,400
Volume: BTUs per cubic foot
Natural gas delivers about 1,030 BTU per cubic foot (the exact value varies by supplier). A furnace burns 150 ft³ of gas on a cold evening. How much heat is released?
Equation: 1,030 × 150 = 154,500 BTU
Volume: mass density
An aluminum rod is a cylinder 4 cm across and 50 cm long. Aluminum has a density of 2.70 g/cm³. Find the mass of the rod.
Equation: V = π(2²)(50) = 200π ≈ 628.3 cm³; mass = 2.70 × 628.3 ≈ 1,696 g ≈ 1.70 kg
Area: rate per area in a design
Grass seed is spread at 5 lb per 1,000 ft². A circular lawn has a radius of 30 ft. How many 5-lb bags are needed?
Equation: Area = 900π ≈ 2,827 ft²; seed = 5 × 2.827 ≈ 14.1 lb, so 3 bags
After the examples, work Diagram 2 together: heating the air in a room is a density based on volume, because the heat needed is proportional to the volume of air. Then address units, where many errors happen. Write the conversions: 1 mi² = 640 acres, 1 ft = 12 in so 1 ft³ = 1,728 in³, 1 yd³ = 27 ft³, and 1 g/cm³ = 1,000 kg/m³. Stress that square and cubic units convert with the square and the cube of the length factor.
Guided Practice15 minutes
Pairs solve three problems and label each as area-based or volume-based before they start:
An invented town of 24,300 people covers 9 square miles: 24,300 ÷ 9 = 2,700 persons per square mile (area).
A copper cube 3 cm on each side has a mass of 241.92 g: 241.92 ÷ 27 = 8.96 g/cm³ (volume).
A rectangular pool 8 m by 4 m, filled to a depth of 1.5 m, holds 48 m³ of water. At 1,000 kg/m³, the water has a mass of 48,000 kg (volume).
Listen for students who divide the wrong way (area ÷ population), who forget to cube the side of the cube, and who mix centimeters and meters.
Independent Practice10-15 minutes
Students work alone on four problems:
A national park of 1,200 square miles has an elk herd of 3,000: 2.5 elk per square mile.
A granite countertop is 3 m by 0.6 m by 3 cm, and granite has a density of about 2,700 kg/m³: V = 0.054 m³, mass ≈ 145.8 kg.
A garage 24 ft by 22 ft by 9 ft holds 4,752 ft³ of air. At 0.018 BTU per ft³ per °F, warming it by 15°F takes about 1,283 BTU.
A 40-acre field is seeded at 120 lb per acre: 4,800 lb of seed.
Closure5-10 minutes
Exit ticket: (1) An invented town has 45,000 people on 12.5 square miles. Find its density. (Answer: 3,600 persons per square mile.) (2) Explain in one sentence why BTUs per cubic foot is a density based on volume and persons per square mile is a density based on area. (3) Write the relationship density = amount ÷ size in the form that finds the amount.
Differentiation Strategies
For Struggling Students
Give a density triangle with amount on top and density and size on the bottom, and have students cover the unknown
Have students write units on every number and cancel them, so a wrong division shows up as wrong units
Start with rectangles and rectangular prisms before circles, triangles and cylinders
For Advanced Students
Compare the population density of two invented regions when one is modeled with a circle and the other with a triangle, then find what radius would make them equal
Estimate the mass of air in the classroom using 1.2 kg per cubic meter, and explain why it is more than most students expect
Find the number of cubic feet of natural gas needed to heat a room by 20°F if the furnace delivers only 80% of the gas's energy to the air
Assessment Guidance
What to Look For
Look for correct units on every answer (persons per mi², g/cm³, BTU), since the units show whether the student divided in the right order. Check that the geometry step is right: a circle's area uses the radius squared, a cube's volume uses the side cubed. In conversion problems, watch for students who multiply by 12 instead of 144 or 1,728. Strong answers also judge reasonableness, for example that a wooden block with density 0.6 g/cm³ floats.
02
Classroom Activities
3 Activities
1
How Many People Fit?
20 minWhole class, then groups of 3-4
Students measure crowd density with their own bodies. They tape a 2 m by 2 m square on the floor, count how many people fit at different comfort levels, and use the density to estimate the capacity of a larger space.
Procedure
Tape a 2 m by 2 m square (4 m²) on the floor with masking tape and meter sticks
Count how many students stand comfortably inside with room to move. Sample result: 6 students, so 6 ÷ 4 = 1.5 persons per m²
Repeat for a tightly packed line, and record the second density
Groups use each density to estimate how many people fit in a school plaza 40 m by 25 m (1,000 m²): 1,500 people at 1.5 persons per m²
Discussion Questions
Why did we measure a small area and scale up, instead of counting a crowd in the plaza?
Which parts of the plaza (benches, planters) should be subtracted from its area before using the density?
Why do event planners care about people per square meter and not just about the number of people?
Modification for Distance Learning
Students mark 1 m by 1 m on the floor at home with tape or string, place standing cutouts or chairs to represent people, and share photos. The class pools the results and compares densities.
2
Density Detectives
20 minGroups of 3
Groups find the mass density of a regular solid, where geometry gives the volume, and of an irregular object, where water displacement gives it. They compare their values with known densities and decide what each object might be made of.
Procedure
Measure a wooden block with a ruler and weigh it. Sample: 10 cm by 5 cm by 4 cm = 200 cm³ and 100 g, so 0.5 g/cm³, a typical value for pine
Put 50 mL of water in a graduated cylinder, drop in a small stone and read the new level. Sample: 72 mL, so the stone's volume is 22 cm³. With a mass of 59.4 g, its density is 2.7 g/cm³
Compare with water, 1 g/cm³: predict whether each object floats, and test it
Discussion Questions
Why do we need geometry for the block but not for the stone?
Your block measurements are to the nearest millimeter. How much could that change the density?
If you cut the block in half, what happens to its mass, its volume and its density?
3
Which Neighborhood Is Most Crowded?
20 minPairs
Pairs receive a map of four invented neighborhoods, each modeled by a simple shape, with a population for each. They compute each area with geometry, find the densities, and rank the neighborhoods.
Neighborhood Data (Invented)
Northside, a rectangle 2.5 mi by 1.6 mi (4 mi²), population 18,400: 4,600 persons per mi²
Riverside, a right triangle with legs 3 mi and 2 mi (3 mi²), population 15,900: 5,300 persons per mi²
Hilltop, a circle of radius 1 mi (π ≈ 3.14 mi²), population 12,000: about 3,820 persons per mi²
Old Town, a square 1.2 mi on a side (1.44 mi²), population 9,000: 6,250 persons per mi²
Procedure
Compute each area, then each density, and rank the neighborhoods from most to least crowded (Old Town, Riverside, Northside, Hilltop)
Compare the ranking by population with the ranking by density and explain why they differ
The city plans one new bus line for the most crowded neighborhood. Write a two-sentence recommendation using your numbers
Challenge Variation
Hilltop expects 4,000 new residents. What radius would its circle need so that its density stays the same? (About 1.15 mi, since the area must grow by a factor of 16,000 ÷ 12,000.)
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Same Population, Different Densities
Two invented towns, drawn to the same scale, each with 24,000 people (one dot = 1,000 people). Town A packs them into 12 square miles, 2,000 people per square mile; Town B spreads them over 24 square miles, 1,000 people per square mile. Density based on area compares places of different sizes.
Diagram 2: Heating a Room, a Density Based on Volume
A room 20 ft by 15 ft by 8 ft holds 2,400 ft³ of air. Warming air takes about 0.018 BTU per cubic foot for each degree Fahrenheit, so a 10°F rise takes 432 BTU, the heat in about 0.42 ft³ of natural gas at 1,030 BTU per cubic foot, ignoring losses. The width and height are to scale; the depth is drawn at half scale in an oblique view.
04
Homework Assignment
~30 min
HSG.MG.A.2 Homework: Density Based on Area and Volume
Directions: For each problem, say whether the density is based on area or on volume. Show the area or volume you compute, write units on every number, and round sensibly for the context.
Part 1: Density Based on Area (Problems 1-3)
An invented state has an area of 84,000 square miles and a population of 2,352,000. Find its population density.
A farm pond is roughly a circle 50 m across. The owner wants no more than 1 fish for every 4 m² of surface. Model the pond as a circle and find the largest number of fish the pond should hold.
An invented town has 9,800 people on 3.5 square miles. Find its density in persons per square mile and in persons per acre (1 mi² = 640 acres).
Part 2: Density Based on Volume (Problems 4-6)
A concrete patio slab is 12 ft by 10 ft and 4 inches thick. Concrete weighs about 150 lb per cubic foot. (a) Find the volume in cubic feet and the weight of the slab. (b) Concrete is ordered in cubic yards. How many cubic yards is the slab?
A gas water heater is rated at 40,000 BTU per hour. Natural gas delivers about 1,030 BTU per cubic foot. (a) How many cubic feet of gas does it burn in one hour? (b) The gas company charges $1.60 per 100 ft³. What does 3 hours of heating cost?
An oak log is a cylinder 0.5 m in diameter and 3 m long. Use 750 kg per cubic meter for the density of oak. Find the mass of the log and decide whether two people could lift it.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Area or Volume
Correct geometric model and area or volume computed
Correct model with one error in the formula
Area or volume missing or wrong
Density Relationship
Correct form used (divide or multiply) with units that cancel
Correct form, units missing
Division in the wrong order
Unit Conversions
Square and cubic conversions correct
One conversion error
Conversions missing or wrong
Interpretation
Answer rounded for the context and judged for reasonableness
Correct value, no interpretation
No conclusion
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
An invented city has 600,000 residents on 150 square miles. What is its population density?
Answer: C
Density = people ÷ area = 600,000 ÷ 150 = 4,000 persons per square mile. Choice A multiplies instead of dividing. Choice B divides area by people, which gives square miles per person. Choice D is off by a factor of 10.
Question 2 of 20 · Multiple Choice
Which of these is a density based on volume?
Answer: B
A liter is a unit of volume, so grams of salt per liter is a volume-based density. Persons per square mile and bushels per acre are based on area, and miles per hour is a rate over time, not a density.
Question 3 of 20 · Multiple Choice
A metal block is 2 cm by 3 cm by 5 cm and has a mass of 237 g. What is its density?
Answer: A
V = 2 × 3 × 5 = 30 cm³, so density = 237 ÷ 30 = 7.9 g/cm³, close to the density of iron. Choice B divides by the sum of the dimensions, 10. Choice C divides volume by mass. Choice D multiplies mass by volume.
Question 4 of 20 · Multiple Choice
An invented county has a density of 250 persons per square mile and an area of 480 square miles. About how many people live there?
Answer: D
Amount = density × area = 250 × 480 = 120,000 people. Choice A divides 480 by 250, and choice C divides 250 by 480. Choice B adds the two numbers.
Question 5 of 20 · Multiple Choice
An invented town is modeled as a circle with a radius of 2 miles and has 50,000 residents. What is its population density, to the nearest whole number?
Answer: A
Area = π(2)² = 4π ≈ 12.57 mi², so density = 50,000 ÷ 4π ≈ 3,979. Choice B uses πr = 2π as the area, forgetting to square the radius. Choice C divides by r² without π. Choice D uses 4 as the radius.
Question 6 of 20 · Multiple Choice
An aluminum ball has a radius of 3 cm. Aluminum has a density of 2.7 g/cm³. What is the mass of the ball, to the nearest tenth?
Answer: C
V = ⁴⁄₃π(3)³ = 36π ≈ 113.1 cm³, and mass = 2.7 × 113.1 ≈ 305.4 g. Choice A stops at the volume. Choice B divides the volume by the density. Choice D squares the radius instead of cubing it.
Question 7 of 20 · Multiple Choice
An invented town of 12,800 people covers 5 square miles. How many people is that per acre? (1 mi² = 640 acres)
Answer: B
5 mi² = 5 × 640 = 3,200 acres, so the density is 12,800 ÷ 3,200 = 4 persons per acre. Choice A is persons per square mile. Choice C divides by 640 only and ignores the 5. Choice D divides acres by people.
Question 8 of 20 · Multiple Choice
Natural gas delivers about 1,030 BTU per cubic foot. How much heat do 500 ft³ of gas release?
Answer: D
Heat = density × volume = 1,030 × 500 = 515,000 BTU. Choice A divides the volume by the density, and choice C divides the density by the volume. Choice B adds the two numbers.
Question 9 of 20 · Multiple Choice
A cylindrical water tank has a radius of 1 m and a height of 2 m. Water has a density of 1,000 kg/m³. What is the mass of the water in a full tank, to the nearest kilogram?
Answer: A
V = π(1)²(2) = 2π ≈ 6.283 m³, so mass = 1,000 × 2π ≈ 6,283 kg. Choice B leaves out π. Choice C uses the height 2 as the radius. Choice D stops at the volume in cubic meters and drops the density.
Question 10 of 20 · Multiple Choice
A student divides 30 square miles by 90,000 people and gets about 0.00033. What does this number measure?
Answer: C
Square miles divided by people gives square miles per person, the reciprocal of population density. It has a meaning (the average area for each person), but it is not persons per square mile, which is 90,000 ÷ 30 = 3,000. Writing units on both numbers shows which quantity you computed.
Question 11 of 20 · Multiple Choice
Invented Town A has 20,000 people on 8 square miles. Invented Town B has 30,000 people on 15 square miles. Which is more densely populated?
Answer: A
Town A: 20,000 ÷ 8 = 2,500 persons per mi². Town B: 30,000 ÷ 15 = 2,000 persons per mi². Town A is denser by 500. Choice B compares totals instead of densities, the exact mistake density is meant to prevent.
Question 12 of 20 · Multiple Choice
A rectangular field is 400 m by 250 m. A farmer plants corn at 8 plants per square meter. About how many plants is that?
Answer: B
Area = 400 × 250 = 100,000 m², and 8 × 100,000 = 800,000 plants. Choices A and C are off by a factor of 10. Choice D divides the area by 8 instead of multiplying.
Question 13 of 20 · Multiple Choice
Glass has a density of about 2.5 g/cm³. What is that in kilograms per cubic meter?
Answer: D
1 m³ = 100³ = 1,000,000 cm³ and 1 kg = 1,000 g, so 1 g/cm³ = 1,000,000 ÷ 1,000 = 1,000 kg/m³. Then 2.5 g/cm³ = 2,500 kg/m³. Choice C multiplies by 100, the length factor alone, instead of by 100³ ÷ 1,000 = 1,000.
Question 14 of 20 · Multiple Choice
An installer's guideline for a room is 20 BTU per hour of cooling for each square foot of floor. What cooling does a room 18 ft by 14 ft need?
Answer: C
Floor area = 18 × 14 = 252 ft², so the room needs 20 × 252 = 5,040 BTU per hour. This density is based on area. Choice A adds the side lengths before multiplying. Choice B divides the area by 20. Choice D multiplies by a 9-ft ceiling as if the guideline were per cubic foot.
Question 15 of 20 · Short Answer
An invented town is modeled as a right triangle with legs of 4 miles and 3 miles. It has 21,000 residents. Find its population density.
Area = ½(4)(3) = 6 mi². Density = 21,000 ÷ 6 = 3,500 persons per square mile. Forgetting the ½ would give 1,750, half the correct value.
Question 16 of 20 · Short Answer
A block of wood is 20 cm by 10 cm by 5 cm and has a mass of 600 g. Find its density. Will it float in water, which has a density of 1 g/cm³?
V = 20 × 10 × 5 = 1,000 cm³. Density = 600 ÷ 1,000 = 0.6 g/cm³. That is less than 1 g/cm³, so the block floats.
Question 17 of 20 · Short Answer
A home used 7,500 ft³ of natural gas in January. At 1,030 BTU per cubic foot, how many BTUs is that? Gas bills often use therms, where 1 therm = 100,000 BTU. How many therms did the home use?
One inch of rain falls on a flat roof that measures 1,500 ft². Water weighs about 62.4 lb per cubic foot. What is the weight of the water that falls on the roof?
Volume = area × depth = 1,500 × (1/12) = 125 ft³. Weight = 62.4 × 125 = 7,800 lb. The area gives the volume, and the volume-based density gives the weight.
Question 19 of 20 · Short Answer
An invented island is modeled as a circle with a radius of 5 km. Its population density is 120 persons per square kilometer. About how many people live on the island?
Area = π(5)² = 25π ≈ 78.5 km². Population = 120 × 25π = 3,000π ≈ 9,425 people.
Question 20 of 20 · Short Answer
A concert crowd fills a rectangular field 110 m by 70 m at about 2 persons per square meter. Estimate the size of the crowd, and name one reason the estimate might be too high.
Area = 110 × 70 = 7,700 m², so the crowd is about 2 × 7,700 = 15,400 people. It may be too high because the stage, aisles and equipment take up part of the field, and the crowd is usually less dense at the back.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSG.MG.A.2 mean?
HSG.MG.A.2 means students apply density, an amount per unit of area or per unit of volume, to model real situations. The official examples are persons per square mile, a density based on area, and BTUs per cubic foot, a density based on volume. Geometry provides the area or volume in each problem.
Is HSG.MG.A.2 about mass density from science class?
Partly. Mass density in grams per cubic centimeter is one example of a density based on volume, but HSG.MG.A.2 is broader. It also includes population density, energy per cubic foot of fuel, and any amount spread over an area, such as seed per square foot or trees per acre.
What is the formula for density?
Density = amount ÷ size, where the size is an area or a volume. The same relationship gives amount = density × size and size = amount ÷ density. The units tell you which form you need: persons per square mile times square miles gives persons.
What is a BTU?
A British thermal unit is a unit of heat energy: the heat needed to warm one pound of water by one degree Fahrenheit. Fuels such as natural gas are described by their BTUs per cubic foot, and heaters and air conditioners by their BTUs per hour.
What are common mistakes with density problems?
A common error is dividing in the wrong order, such as area ÷ people. Others are converting square or cubic units with the length factor instead of its square or cube (multiplying by 12 instead of 144), using a diameter as a radius when finding the area of a circular region, and comparing total amounts when the question is about density.
How do you convert square miles to acres or cubic feet to cubic yards?
1 square mile = 640 acres, so divide persons per square mile by 640 to get persons per acre. 1 yd = 3 ft, so 1 yd³ = 3³ = 27 ft³. The general rule: a square unit converts with the square of the length factor, and a cubic unit with its cube.
Why does HSG.MG.A.2 belong to geometry?
Because the size in every density problem is an area or a volume, and in modeling situations that size comes from a geometric model: a region as a rectangle or circle, a tank as a cylinder, a room as a prism. The standard is in the Modeling with Geometry domain for that reason.
How is HSG.MG.A.2 tested?
Typical items give a region or an object with its dimensions and a total or a density, and ask for the missing quantity, often with a unit conversion. Some ask students to compare two places by density or to judge whether an answer is reasonable.
How does this standard connect to other courses?
It builds on unit rates from Grades 6 and 7 (6.RP.A.2) and on area and volume formulas. It connects to chemistry and physics (mass density), to environmental science and geography (population and wildlife density), and to engineering, where energy and material densities drive design choices (HSG.MG.A.3).
Is density on the SAT?
Yes. Rates and density appear in the Problem-Solving and Data Analysis domain of the digital SAT Math section, and area and volume in the Geometry and Trigonometry domain. Problems that combine a volume with a density use both.
07
Related Standards
6 standards
These standards connect to HSG.MG.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.RP.A.2Prerequisite
Understand a unit rate a/b for a ratio a:b and use rate language
