Grade 6 Workbook — Chapter 13: Area and Perimeter — Triangles and Parallelograms

SOL 6.MG.2 · Companion to Textbook Chapter 13

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/.

PAGE 1 — Chapter opener

Chapter 13 · Area and Perimeter: Triangles and Parallelograms

Standard 6.MG.2

In this chapter you will:

• Find the perimeter of triangles and parallelograms
• Build the parallelogram area formula by cutting and rearranging
• Build the triangle area formula from two congruent triangles
• Solve area and perimeter problems in real situations

Words to know: perimeter · area · base · adjacent side · height · congruent · square units · linear units

Materials for this chapter: grid paper, scissors, a ruler. You will cut shapes apart on purpose.

PAGE 2 — Perimeter is the distance around

13.1 Perimeter

FIGURE: fig1-parallelogram-parts.png (right half of page)

[A leaning parallelogram with the bottom side labeled "base (b)", the slanted left side labeled "adjacent side (s)", and a dashed perpendicular segment from the top-left vertex to the base labeled "height (h)" with a right-angle mark. Tick marks show the congruent pairs of sides. Caption: "the height is not a side of the figure."]

Triangle: $P = a + b + c$ Parallelogram: $P = 2(b + s)$ — opposite sides are congruent Perimeter never uses the height.

Find each perimeter. Include units.

PAGE 3 — Work backward

Missing Side Lengths

FIGURE: fig2-perimeter-sum-of-sides.png (full width)

[Two worked figures side by side: a triangle with sides 7 cm, 9 cm, 12 cm above the line "P = 7 + 9 + 12 = 28 cm"; a leaning parallelogram with bottom 14 in and slanted side 9 in, congruent pairs labeled in lighter type, above the line "P = 2(14 + 9) = 46 in".]

Find the missing side. Show your check.

1. A parallelogram has perimeter 48 in and base 15 in. Adjacent side = ______

   Check: ______________________

2. A triangle has perimeter 31 cm with sides 9 cm and 12 cm. Third side = ______

   Check: ______________________

3. A parallelogram has perimeter 40 ft and base 12 ft. Adjacent side = ______

   Check: ______________________

Cross out the number you do NOT need. A parallelogram has base 9 m, height 4 m, and adjacent side 5 m. Find the perimeter.

Perimeter = ______ Unused measurement = ______ Why? _______________________

PAGE 4 — Exit ticket 13.1

Exit Ticket · Lesson 13.1

Name: ________________________ Date: ____________

1. Perimeter of a triangle with sides 7 m, 10 m, 13 m: ______

2. Perimeter of a parallelogram with base 11 in and adjacent side 6 in: ______

3. A parallelogram has perimeter 40 ft and base 12 ft. Adjacent side: ______

4. Why do you never need the height to find a perimeter?

PAGE 5 — Count the squares

13.2 Building the Parallelogram Formula

FIGURE: fig3-rectangle-area-grid.png (full width)

[A rectangle on a square grid, 6 squares wide and 4 squares tall with faint interior grid lines. Bottom edge labeled "base = 6 units," left edge labeled "height = 4 units." One row tinted and labeled "6 squares in each row," a left bracket labeled "4 rows." Caption: "6 x 4 = 24 square units."]

Area = the number of square units a figure covers.

Count or multiply to find each rectangle's area.

Rectangle formula: $A =$ ______ $\times$ ______

PAGE 6 — Cut and slide

From Parallelogram to Rectangle

FIGURE: fig4-parallelogram-cut-and-slide.png (full width)

[Three panels on a light grid joined by arrows labeled "cut" and "slide." Panel 1: a leaning parallelogram, base labeled b, dashed perpendicular height labeled h with a right-angle mark, the small right triangle at the left end tinted. Panel 2: the two pieces separated along the dashed line. Panel 3: a rectangle with the tinted triangle fitted onto the right end, base b and height h labeled. Caption: "same base, same height, same area."]

Do it yourself. Draw a parallelogram on grid paper. Draw the height. Cut along it. Slide the triangle to the other end.

What shape do you get? _______________________

Did the base change? ______ Did the height change? ______ Did the area change? ______

So the parallelogram formula must be: $A =$ ______________

Explain. Why does moving a piece leave the area unchanged?

PAGE 7 — Height or side?

The Height Is Not a Side

FIGURE: fig5-height-vs-side.png (full width)

[The same leaning parallelogram drawn twice with identical dimensions. Left copy: slanted side bold, labeled "side = 5 ft (use for PERIMETER)." Right copy: dashed perpendicular from the top-left vertex to the base bold, labeled "height = 3 ft (use for AREA)" with a right-angle mark, base labeled 10 ft. Below: "P = 2(10 + 5) = 30 ft" and "A = 10 x 3 = 30 square ft," with the note "the side and the height are different lengths."]

The height meets the base at a RIGHT ANGLE. Look for the small square mark.

Underline the height in each list of measurements. Then find the area.

Find the missing measurement.

$A = 72$ cm², base 8 cm. Height = ______

$A = 90$ m², height 6 m. Base = ______

$A = 63$ m², base 7 m. Height = ______

Explain. Why is a parallelogram's slanted side always longer than its height?

PAGE 8 — Exit ticket 13.2

Exit Ticket · Lesson 13.2

Name: ________________________ Date: ____________

1. Area of a parallelogram with base 14 in and height 6 in: ______

2. A parallelogram has area 63 m² and base 7 m. Height: ______

3. A diagram shows a slanted side of 8 cm and a dashed segment of 5 cm meeting the base at a right angle. Which is the height? ______

4. How does rearranging a parallelogram into a rectangle show that $A = bh$?

PAGE 9 — Two triangles make a parallelogram

13.3 Building the Triangle Formula

FIGURE: fig6-two-triangles-make-parallelogram.png (full width)

[Two panels joined by an arrow labeled "copy and rotate a half turn." Left: a scalene triangle, base labeled b, dashed perpendicular height labeled h with a right-angle mark. Right: a parallelogram made of the original triangle plus a tinted congruent copy rotated a half turn and joined along the slanted side, with base b and dashed height h labeled. Caption: "the triangle is exactly half of the parallelogram."]

Do it yourself. Cut out two congruent triangles. Rotate one a half turn and join them along a side.

What shape do the two triangles make? _______________________

The parallelogram's area is ______________. One triangle is ______ of it.

So the triangle formula must be: $A =$ ______________

Find each area.

PAGE 10 — Three kinds of height

Where the Height Lands

FIGURE: fig7-triangle-heights-three-cases.png (full width)

[Three triangles in a row, each with a dashed height and a right-angle mark. Left, headed "acute": height falls inside, base labeled b, height labeled h. Middle, headed "right": right angle at bottom left, horizontal leg labeled "base," vertical leg labeled "height," hypotenuse labeled "hypotenuse - never the height." Right, headed "obtuse": height from the top vertex lands outside the base on a dotted extension of the base line, labeled h.]

Right triangles. The two legs are already perpendicular, so they are the base and the height. The hypotenuse is never the height.

Obtuse triangle. Base 6 in, height 9 in falling outside the triangle. Area = ______

Find the missing measurement.

$A = 48$ cm², height 8 cm. Base = ______

$A = 35$ m², base 10 m. Height = ______

$A = 30$ ft², base 10 ft. Height = ______

Error hunt. A student found the area of the 5-12-13 right triangle as $\tfrac{1}{2}(5)(13) = 32.5$ ft². What went wrong?

Correct area: ______

PAGE 11 — Exit ticket 13.3

Exit Ticket · Lesson 13.3

Name: ________________________ Date: ____________

1. Area of a triangle with base 14 m and height 6 m: ______

2. A triangle has area 54 in² and base 12 in. Height: ______

3. A right triangle has legs 3 cm and 8 cm. Area: ______

4. Why is a triangle's area half that of a parallelogram with the same base and height?

PAGE 12 — Area or perimeter?

13.4 Which One Does the Problem Want?

Write "area" or "perimeter" for each job.

Rounding rule. If a panel is 96 ft² and one can of paint covers 80 ft², then $96 \div 80 =$ ______ cans.

You must buy ______ whole cans. Why do you round this way?

PAGE 13 — Problems in context

Real Situations

FIGURE: fig8-context-sail-and-patio.png (full width)

[Two labeled diagrams. Left: a triangular sail with base 8 ft and a dashed perpendicular height 15 ft with a right-angle mark, captioned "area: how much sailcloth." Right: a leaning parallelogram patio with bottom side 14 m, slanted side 10 m, and dashed perpendicular height 9 m with a right-angle mark, captioned "area: how many pavers - perimeter: how much edging."]

1. Triangular garden, base 10 ft, height 6 ft. One bag of soil covers 10 ft².

   Area = ______ Bags needed = ______

2. Patio parallelogram, base 14 m, height 9 m, adjacent side 10 m.

   Area = ______ Edging = ______

3. Triangular sail, base 8 ft, height 15 ft. Sailcloth costs $9 per square foot.

   Area = ______ Cost = ______

4. Triangular sail with area 84 ft² and base 12 ft. Height = ______ Check: ______________

5. Field parallelogram, base 60 yd, height 40 yd, adjacent side 45 yd.

   Area = ______ Fencing = ______

6. Triangular dog run, sides 18 ft, 24 ft, 30 ft, with a 3-ft gate that needs no fence. Fencing costs $4 per foot.

   Perimeter = ______ Length fenced = ______ Cost = ______

PAGE 14 — Compare and decide

Comparing Figures

Which uses less fabric? Triangle A: base 9 in, height 8 in. Triangle B: base 12 in, height 6 in.

Area A = ______ Area B = ______ Conclusion: _______________________

Same base and height. A triangle and a parallelogram both have base 10 cm and height 6 cm.

Triangle area = ______ Parallelogram area = ______ Relationship: _______________________

Same base and height, different perimeters? Two parallelograms each have base 10 cm and height 4 cm, but their slanted sides are 5 cm and 7 cm.

Areas: ______ and ______ Perimeters: ______ and ______

What does this tell you about area and perimeter?

Fabric and ribbon. A banner is a parallelogram with base 5 ft, height 3 ft, and adjacent side 3.5 ft.

Fabric needed = ______ Ribbon needed = ______

Why is the height used for one and not the other?

PAGE 15 — Exit ticket 13.4

Exit Ticket · Lesson 13.4

Name: ________________________ Date: ____________

1. Triangular tabletop, base 6 ft, height 4 ft. Area: ______

2. Parallelogram garden, base 11 m, adjacent side 7 m. Perimeter: ______

3. A parallelogram has area 84 in² and base 12 in. Height: ______

4. How do you decide whether a problem is asking for area or perimeter?

PAGE 16 — Chapter 13 review, part 1

Chapter 13 Review

Part A · Developing the formulas

1. Describe the cut-and-slide move that turns a parallelogram into a rectangle. What stays the same?

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2. How do two congruent triangles show that $A = \tfrac{1}{2}bh$?

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3. Grid paper parallelogram, base 7 units, height 4 units. Area = ______ square units

4. A diagram shows base 10 cm, slanted side 8 cm, and a dashed 6 cm segment meeting the base at a right angle.

   The height is ______ Area = ______

Part B · Perimeter

5. Triangle 13 in, 14 in, 15 in. $P =$ ______

6. Parallelogram base 16 cm, side 9 cm. $P =$ ______

7. Parallelogram with $P = 54$ ft and base 20 ft. Adjacent side = ______

8. Triangle with $P = 40$ m and sides 12 m and 15 m. Third side = ______

PAGE 17 — Chapter 13 review, part 2

Chapter 13 Review (continued)

Part C · Area

9. Parallelogram base 18 m, height 7 m. $A =$ ______

10. Triangle base 20 cm, height 9 cm. $A =$ ______

11. Parallelogram with $A = 96$ in² and height 8 in. Base = ______

12. Triangle with $A = 45$ ft² and base 15 ft. Height = ______

Part D · Problems in context

13. Triangular flower bed, base 12 ft, height 7 ft. One bag of mulch covers 15 ft².

    Area = ______ Bags = ______

14. Deck parallelogram, base 20 ft, height 12 ft, adjacent side 13 ft.

    Decking area = ______ Railing = ______

15. Triangular sail, base 9 ft, height 16 ft, cloth $12 per square foot.

    Area = ______ Cost = ______

16. Parallelogram sign base 10 m, height 6 m. Triangular sign base 15 m, height 8 m.

    Areas: ______ and ______ More to paint: _______________________

PAGE 18 — Chapter 13 review, part 3

Chapter 13 Review (continued)

Part E · Reasoning

17. Why can a parallelogram's slanted side not be used in place of the height when finding area?

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18. Can two parallelograms have the same base and height but different perimeters? Give measurements.

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19. A triangle and a parallelogram have the same base and height. How do their areas compare, and why?

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20. Why is area reported in square units while perimeter is reported in linear units?

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Formula summary — fill it in from memory.

Canva production notes

• Page size: 8.5 × 11 in, 0.75 in margins
• Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
• Materials callout: page 1 lists grid paper and scissors; pages 6 and 9 are hands-on and need clear space beside the figure for taped cut-outs
• Figure widths: all figures full width except fig1-parallelogram-parts.png, which sits at half width on the right of page 2
• Right-angle marks: every height in every figure must carry a visible small square at the base, at least 8 px at print resolution — this is the whole point of the chapter
• Dashed versus solid: heights are dashed, sides are solid, in every figure without exception
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow