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.

Why This Matters Your architecture firm is calculating the floor area of an L-shaped room for a client's new home. The contractor needs the total area to order the correct amount of flooring. The room can be split into two rectangles: one is 12 feet by 8 feet and the other is 6 feet by 5 feet.

Start hereWords for this lesson

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

WordWhat it meansExample
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 example

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

Regular pentagon: sides 10 cm, apothem about 7 cm.

  1. 5 sides make 5 triangles.
  2. One triangle: ½ × 10 × 7 = 35 cm².
  3. Five triangles: 5 × 35 = 175.

Answer: Area ≈ 175 cm²

Now you trySame steps, your turn

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

Now an L-shape made of a 5 ft by 4 ft rectangle and a 3 ft by 2 ft rectangle.

Answer:

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 out!A 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

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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.

Why This Matters Your architecture firm is calculating the floor area of an L-shaped room for a client's new home. The contractor needs the total area to order the correct amount of flooring. The room can be split into two rectangles: one is 12 feet by 8 feet and the other is 6 feet by 5 feet.

How it worksWorked example

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

Regular pentagon: sides 10 cm, apothem about 7 cm.

  1. 5 sides make 5 triangles.
  2. One triangle: ½ × 10 × 7 = 35 cm².
  3. Five triangles: 5 × 35 = 175.

Answer: Area ≈ 175 cm²

  1. 1Circle 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
    Steps to try
    1. The L-shape is already split into two rectangles for you: 10 × 6 and 4 × 3.
    2. Find each rectangle's area separately with A = l × w, then combine — joined shapes mean adding.
    3. Compute 10 × 6 and 4 × 3, then add the two results. Label your answer in sq ft.
    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?
  2. 2Circle 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
    Steps to try
    1. The flower bed is CUT OUT of the patio — that phrase tells you which operation to use.
    2. Find the full patio area (15 × 10) and the cutout area (5 × 4), then remove the cutout.
    3. Compute 15 × 10 and 5 × 4, then subtract the flower bed's area from the patio's area.
    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?
  3. 3Circle 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
    Steps to try
    1. Two rectangles make this L: one is 5 × 3 and the other is 2 × 4.
    2. Joined pieces mean you add their areas after computing each one with length × width.
    3. Find 5 × 3, then 2 × 4, and add the two products together.
    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?
  4. 4Circle 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
    Steps to try
    1. Think of real examples: a window in a wall versus two rooms joined into an L-shape.
    2. Ask for each choice: is material missing from the shape, or are pieces being combined?
    3. Picture a wall with a window: you can't paint the glass, so its area comes OFF the wall's area. Match that idea to a choice.
    Show your work

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.

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help