Grade 6 Mathematics · Unit 5: Solve Area, Surface Area, and Volume ProblemsStandard 6.GR.1

Lesson 5.4Apply Area Concepts to Solve Problems

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can find the area of a composite figure by decomposing it into basic shapes, including a regular polygon split into triangles, and adding or subtracting the areas.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

Word	What it means	Example
Composite FigureSpanish: Figura compuesta	A shape made by putting two or more simple shapes together.	triangles joined together
DecomposeSpanish: Descomponer	To break a shape into smaller, simpler shapes.	hexagon → 6 triangles
AddSpanish: Sumar	Add up the areas of the smaller shapes to get the total.	T-shaped hallway: top rectangle = 30 sq ft, bottom rectangle = 28 sq ft → total = 30 + 28 = 58 sq ft
SubtractSpanish: Restar	Take away the area of a missing piece from a bigger shape.	A 14×10 pool with a 6×4 cutout: 140 − 24 = 116 sq ft of water surface
FormulaSpanish: Fórmula	A math rule written with symbols.	Rectangle: A = l × w; Triangle: A = ½ × b × h; use the right formula for each piece of a composite figure
Regular polygonSpanish: Polígono regular	A shape whose sides are all the same length and whose angles are all the same size.	stop sign, hexagon tile

How it worksWorked examplearea of a regular polygon

These numbers are not on your problems. The steps are. Follow them with your own numbers.

A regular octagon has sides of 12 in. A triangle drawn from the center to one side has a height of 14 in. Find the area.

A regular octagon has eight equal sides, so it decomposes into eight congruent triangles.
Find the area of ONE triangle. ½ × 12 × 14 = 84
Every triangle matches, so multiply by how many there are. 84 × 8 = 672

Answer: The area of the octagon is 672 square inches (672 in²).

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

A regular hexagon has sides of 20 m. Each triangle drawn from the center has a height of 17 m. Find the area.

How many congruent triangles does a regular hexagon decompose into?
Area of ONE triangle. ½ × × =
Multiply by the number of triangles. × =

Answer:area of the hexagon

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

I decomposed the figure into congruent .
The area of one triangle is square units.
There are triangles, so the total area is .
I know the triangles are congruent because .

Word bank regular polygondecomposecongruenttrianglecenterbaseheightmultiplytotalareasquare unitscomposite figure

Watch outA common mistake

A common mistake in Area of Composite Figures is adding every piece even when a piece has been cut out of the figure. For a 10 ft by 8 ft rectangle with a 3 ft by 2 ft rectangular notch cut from one corner, students compute the full rectangle 10 × 8 = 80 sq ft and stop, or they add the notch instead of removing it (80 + 6 = 86 sq ft) instead of subtracting it: 80 − (3 × 2) = 80 − 6 = 74 sq ft. Before submitting, check whether each piece is part of the figure (add it) or missing from the figure (subtract it).

Grade 6 Mathematics · Unit 5: Solve Area, Surface Area, and Volume ProblemsStandard 6.GR.1

Lesson 5.4Apply Area Concepts to Solve Problems

Version ASupported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice

Name Date Period

Learning target I can find the area of a composite figure by decomposing it into basic shapes, including a regular polygon split into triangles, and adding or subtracting the areas.

Part 1Understand the ideamodels and meaning

1Write the name of the correct group on each line.
Sort each composite figure by whether you ADD or SUBTRACT the areas to find the total.

Groups: Add the areas Subtract the areas
- L-shaped hallway (two rectangles joined)
- Wall with a window cut out
- T-shaped stage (two rectangles joined)
- Floor with a trapdoor removed
- U-shaped pool (rectangle with center removed)
- Plus-sign shape (rectangles joined)
Hint Look for action words on each card: 'joined' versus 'cut out' or 'removed'.

Part 2Apply itreal-world problems

2Circle the letter of the best answer. Show how you know.
An L-shaped room is made of two rectangles: 10 ft × 6 ft and 4 ft × 3 ft. What is the total area?
1. A12 sq ft
2. B60 sq ft
3. C72 sq ft
4. D72 ft
Hint The L-shape is already split into two rectangles for you: 10 × 6 and 4 × 3.
Formula

Put the numbers in

Work it out

Answer with its unit
1. 1Name itWhich measure is asked?
2. 2Write the formulaBefore any numbers.
3. 3SubstituteMatch each letter.
4. 4Unitunits, sq units, or cubic?
3Circle the letter of the best answer. Show how you know.
A rectangular patio is 15 ft × 10 ft with a 5 ft × 4 ft rectangular flower bed cut out. What is the remaining area?
1. A20 sq ft
2. B130 sq ft
3. C150 sq ft
4. D170 sq ft
Hint The flower bed is CUT OUT of the patio — that phrase tells you which operation to use.
Formula

Put the numbers in

Work it out

Answer with its unit
1. 1Name itWhich measure is asked?
2. 2Write the formulaBefore any numbers.
3. 3SubstituteMatch each letter.
4. 4Unitunits, sq units, or cubic?
4Circle the letter of the best answer. Show how you know.
Find the area of an L-shape made of a 5×3 rectangle and a 2×4 rectangle.
1. A8 sq units
2. B15 sq units
3. C20 sq units
4. D23 sq units
Hint Two rectangles make this L: one is 5 × 3 and the other is 2 × 4.
Formula

Put the numbers in

Work it out

Answer with its unit
1. 1Name itWhich measure is asked?
2. 2Write the formulaBefore any numbers.
3. 3SubstituteMatch each letter.
4. 4Unitunits, sq units, or cubic?
5Circle the letter of the best answer. Show how you know.
When should you subtract areas instead of adding them to find the area of a composite figure?
1. AWhen the shapes overlap
2. BWhen one shape is larger than the other
3. CAlways subtract — never add
4. DWhen a piece is cut out or removed from a larger shape
Hint Think of real examples: a window in a wall versus two rooms joined into an L-shape.
Show your work

Part 3Explain and correctreasoning

6Find the mistake. Explain it, then show the correct work.
Spot the Common Mistake
1. 1Identify the shapesAn L-shaped room is made of two joined rectangles: 8 ft × 5 ft and 4 ft × 3 ft.
2. 2Find each areaFirst rectangle = 8 × 5 = 40 sq ft. Second rectangle = 4 × 3 = 12 sq ft.
3. 3Combine the areasTotal = 40 × 12 = 480 sq ft.
Which step has the mistake? Explain what went wrong, then show the correct work.

Hint Look at the last step. When two pieces are joined, do you add or multiply their areas?
Correct work
Correct answer:

Explain your thinkingFor the U-shaped pool, why did we subtract instead of add? When do you add areas and when do you subtract?

Sentence starter I ___ the areas because the composite figure is made by ___ a piece from a larger shape, so the total area is ___ - ___ = ___ square feet.

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help

Grade 6 Mathematics · Unit 5: Solve Area, Surface Area, and Volume ProblemsStandard 6.GR.1

Lesson 5.4Apply Area Concepts to Solve Problems

Version BCore practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice

Name Date Period

Learning target I can find the area of a composite figure by decomposing it into basic shapes, including a regular polygon split into triangles, and adding or subtracting the areas.

Part 1Understand the ideamodels and meaning

1Complete the table. Show how you found each value.

Complete the table.

Figure	Shape 1 Area	Shape 2 Area	Operation
L-shaped deck	48 sq ft	20 sq ft	Add
Floor with alcove	90 sq ft	15 sq ft	Add
Wall with window cutout	120 sq ft	18 sq ft	Subtract
Courtyard with fountain cutout	200 sq ft	36 sq ft	Subtract

Show your work

2Write the letter of the matching item on each line.
Match each composite figure to its total area.
1. L-shape: 10×6 + 4×3
2. T-shape: 14×4 + 6×10
3. Wall 15×10 minus window 5×4
4. Pool 14×10 minus cutout 6×5
5. L-shape: 8×5 + 3×6
6. Floor 12×8 minus vent 2×3
- A72 sq ft
- B116 sq ft
- C130 sq ft
- D110 sq ft
- E58 sq ft
- F90 sq ft
3Use the model to solve. Show your work.

Wall area = 120 sq ft → painted area (108) + window cutout (12)
Show your work
Answer:
4Write the name of the correct group on each line.
Calculate each composite figure's area. Sort by whether the area is more than 100 sq ft or not.

Groups: Area > 100 sq ft Area ≤ 100 sq ft
- L-shape: 12×8 + 6×5
- T-shape: 10×3 + 4×7
- Wall 15×10 − window 5×4
- L-shape: 9×7 + 3×4
- Floor 20×8 − vent 6×3
- U-shape: 10×8 − 4×3

Part 2Explain and correctreasoning

5Find the mistake. Explain it, then show the correct work.
Spot the Common Mistake
1. 1Identify the shapesA 16 ft × 9 ft floor has a 4 ft × 5 ft supply closet cut out of one corner. Find the usable floor area.
2. 2Find the large rectangle areaA = 16 × 9 = 144 sq ft.
3. 3Find the cutout areaA = 4 × 5 = 20 sq ft.
4. 4Find the usable areaUsable area = 144 + 20 = 164 sq ft.
Which step has the mistake? Explain what went wrong, then show the correct work.

Correct work
Correct answer:

Explain your thinkingFor the U-shaped pool, why did we subtract instead of add? When do you add areas and when do you subtract?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help

Grade 6 Mathematics · Unit 5: Solve Area, Surface Area, and Volume ProblemsStandard 6.GR.1

Lesson 5.4Apply Area Concepts to Solve Problems

ChallengeExtension

Mastery check☐ Exceeds☐ Meets target☐ Needs practice

Name Date Period

Learning target I can find the area of a composite figure by decomposing it into basic shapes, including a regular polygon split into triangles, and adding or subtracting the areas.

Part 1Understand the ideamodels and meaning

1Complete the table. Show how you found each value.

This composite figure is a house shape: a rectangle topped with a triangle. Find each area and the total.

Part Shape Dimensions Area
Walls Rectangle 10 ft × 8 ft
Roof Triangle b = 10 ft, h = 4 ft
Total — —

Show your work

Part	Shape	Dimensions
Walls	Rectangle	10 ft × 8 ft
Roof	Triangle	b = 10 ft, h = 4 ft
Total	—	—

Part 2Explain and correctreasoning

2Find the mistake. Explain it, then show the correct work.
Find the Area Error
1. 1Identify the shapesRectangle: 16 ft × 10 ft, with a 4 ft × 6 ft piece cut from the corner
2. 2Find the large rectangle areaA = 16 × 10 = 160 sq ft
3. 3Find the cutout areaA = 4 × 6 = 24 sq ft
4. 4Find total areaTotal = 160 + 24 = 184 sq ft
Which step has the mistake? Explain what went wrong, then show the correct work.

Correct work
Correct answer:
3Answer in complete sentences.

A T-shaped room can be decomposed in two different ways. Show both ways and prove they give the same total area. Use a T-shape where the top is 14 ft × 4 ft and the stem is 6 ft × 10 ft.
Write your answer

Explain your thinkingFor the U-shaped pool, why did we subtract instead of add? When do you add areas and when do you subtract?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help