📋 Mission Brief
6.G.A.1
Find the area of triangles, parallelograms, rhombuses, and polygons by composing or decomposing shapes.
🗺️ Terrain Survey
Identify the Shapes — Click to Reveal Field Notes
Parallelogram
Opposite sides parallel
Rhombus
All sides equal
Triangle
Three sides
📚 Field Glossary
| Term | Meaning | Example | Key Fact |
|---|---|---|---|
| Area | Number of square units inside a shape | How many 1×1 tiles fit inside? | Always in square units (cm², m², ft²) |
| Base (b) | The bottom side — the side the height is measured from | Any side can be the base | You choose which side is the base |
| Height (h) | Perpendicular distance from base to opposite vertex — always 90° | Straight up from base | Height ≠ slant side |
| Perpendicular | Lines that meet at exactly 90° | Height is ALWAYS perpendicular to base | ∟ right angle symbol |
| Diagonal (d) | Line connecting two opposite corners | Rhombus has two diagonals crossing at 90° | Used in rhombus formula |
| Decompose | Break a shape into simpler shapes to find area | Parallelogram = rearranged rectangle | Key strategy for complex shapes |
Parallelogram
A = b × h
base × perpendicular height
Rhombus — two ways
A = b × h
base × perpendicular height
— or —
A = (d₁ × d₂) ÷ 2
half the product of the diagonals
Triangle
A = (b × h) ÷ 2
half of base × height
🟦 Parallelogram Scroll
Formula: A = b × h
A
Area
Total space inside, in square units
b
Base
The bottom side of the shape
h
Height
Straight up at 90° — NOT the slant side
⚠️
The Slant Side Trap
A parallelogram has a slant side and a height. They are different! The height is the dashed gold line — it drops straight down at 90°. The slant side is the red line — it leans. Only the height goes in the formula.
Area = b × h
40
square units
📓 Field Log (TWR)
In A = b × h —
• b stands for
• h stands for (not the slant side!)
• To find area:
📐
Worked Example — Parallelogram
I
Identify the formula
For a parallelogram: A = b × h. Use the perpendicular height, not the slant side.
II
Label dimensions
Base = 12 cm, Height = 7 cm (slant side = 9 cm — ignore it).
b = 12 cm h = 7 cm
III
Calculate
A = 12 × 7 = 84 cm²
IV
Check units
Area is always in square units — cm², m², ft². Never forget the ²!
💎 Rhombus Scroll
Two Formulas — Use Either One
A = b × h
b
Base
Bottom side
h
Height
Straight up at 90°
Use when you know the base and height
A = (d₁ × d₂) ÷ 2
d₁, d₂
Diagonals
Corner to corner, cross at 90°
÷ 2
Halve it
Rhombus = half the diagonal rectangle
Use when you know the two diagonals
Try Formula 1 — b × h
A = b × h
40
square units
Try Formula 2 — Diagonals
A = (d₁ × d₂) ÷ 2
30
square units
📓 Formula Check — Fill in the Blanks
Formula 1 — A = b × h:
• b stands for
• h stands for (must be perpendicular to b)
Formula 2 — A = (d₁ × d₂) ÷ 2:
• d₁ and d₂ are the
• I divide by 2 because
• I choose Formula 1 when I know
• I choose Formula 2 when I know
📐
Worked Examples — Both Rhombus Formulas
1a
Formula 1: A = b × h
A rhombus floor tile has base = 9 cm and perpendicular height = 7 cm.
A = 9 × 7 = 63 cm²
1b
Formula 2: A = (d₁ × d₂) ÷ 2
A baseball diamond (rhombus) has diagonals d₁ = 127 ft and d₂ = 127 ft.
A = (127 × 127) ÷ 2 = 16,129 ÷ 2 = 8,064.5 ft²
⚠️
Which formula to use?
If the problem gives you base and height → use A = b × h.
If the problem gives you the two diagonals → use A = (d₁ × d₂) ÷ 2.
Both give the same answer when applied correctly.
If the problem gives you the two diagonals → use A = (d₁ × d₂) ÷ 2.
Both give the same answer when applied correctly.
🔺 Triangle Scroll
Formula: A = (b × h) ÷ 2
A
Area
Total space inside, in square units
b, h
Base & Height
Base = bottom side. Height = straight up at 90° to the base.
÷ 2
Divide by 2
A triangle is exactly half a parallelogram with the same base and height
Area = (b × h) ÷ 2
30
square units
📓 Field Log (TWR)
In A = (b × h) ÷ 2 —
• b stands for
• h stands for
• I divide by 2 because
📐
Worked Example — Obtuse Triangle (Tricky!)
I
Warning
For an obtuse triangle, the height falls outside the shape. The formula still works identically: A = (b × h) ÷ 2.
II
Label
b = 14 m h = 9 m (outside the triangle)
III
Calculate
A = (14 × 9) ÷ 2 = 126 ÷ 2 = 63 m²
IV
⚠️ Trap to avoid
Never use the slant side as height. Height always forms 90° with the base.
🔗 The Grand Codex — How All Formulas Connect
Every formula connects to the rectangle. A parallelogram is a rearranged rectangle. A triangle is half a parallelogram. A rhombus uses diagonals to find the same half-rectangle relationship.
0
Solved
12
Missions
0%
Score
⛺ Sector Alpha — Single-Step Missions
Missions 1–4 · 5 XP each
1
Parallelogram
+5 XP
📍 Sector Alpha, Site 1: Explorers discover a stone panel shaped like a parallelogram at the edge of camp.
Find the area of the stone panel. Base = 14 cm, Height = 9 cm.
cm²
📜 Field Notes — Show Your Work
Use A = b × h. Multiply the base (14) by the height (9). The slant side is a trap — ignore it!
2
Rhombus
+5 XP
📍 Sector Alpha, Site 2: A kite-shaped rhombus marker is found embedded in the canyon wall.
A rhombus kite has diagonals of 16 m and 10 m. What is its area?
m²
📜 Field Notes — Show Your Work
Rhombus formula: A = (d₁ × d₂) ÷ 2. Multiply 16 × 10 = 160, then divide by 2.
3
Triangle
+5 XP
📍 Sector Alpha, Site 3: A triangular garden plaza is mapped at the expedition's base coordinates.
The triangular plaza has base = 18 ft and height = 11 ft. Find its area.
ft²
📜 Field Notes — Show Your Work
Triangle: A = (b × h) ÷ 2. Multiply 18 × 11 = 198, then divide by 2.
4
Parallelogram — ⚠️ Trap!
+5 XP
📍 Sector Alpha, Site 4: A decoy marker has three measurements — only two are needed. Can you identify the trap?
Base = 13 cm, slant side = 10 cm, height = 7.5 cm. What is the area?
cm²
📜 Field Notes — Show Your Work
Don't use the slant side (10 cm)! Only base × perpendicular height. A = 13 × 7.5
⚔️ Sector Beta — Multi-Step Missions
Missions 5–8 · 8 XP each
5
Missing Dimension
+8 XP
📍 Sector Beta, Site 5: The height measurement on this triangular marker was eroded away. Work backwards to find it.
A triangle has area = 48 in² and base = 12 in. Find the missing height.
in
📜 Field Notes — Show Your Work
Work backwards: 48 = (12 × h) ÷ 2 → multiply both sides by 2: 96 = 12 × h → h = 96 ÷ 12.
📓 Field Log (TWR)
I can work backwards from the formula: if A = 48 and b = 12, then h = because .
6
Real World · Cost
+8 XP
📍 Sector Beta, Site 6: The expedition needs to replace a diamond-shaped window in the field station.
A rhombus window: horizontal diagonal = 2.4 ft, vertical diagonal = 3.6 ft. Replacement costs $18/ft². Total cost?
dollars
📜 Field Notes — Show Your Work
Use A = b × h (base × height) because you're given base and height, not diagonals. Tile area = 11 × 8 = 88 m². Tiles needed = 440 ÷ 88.
7
Missing Dimension
+8 XP
📍 Sector Beta, Site 7: The base stone is missing from this parallelogram marker. Calculate it from the remaining data.
A parallelogram: area = 91 m², height = 7 m. What is the base?
m
📜 Field Notes — Show Your Work
Work backwards: A = b × h → 91 = b × 7 → b = 91 ÷ 7.
8
Composite · Fractions
+8 XP
📍 Sector Beta, Site 8: A triangular rooftop at the dig site needs solar panels installed on part of it.
Triangle roof: base = 24 m, height = 6.5 m. Solar panels cover ⅔ of the roof. Panel area?
m²
📜 Field Notes — Show Your Work
Step 1: Roof area = (24 × 6.5) ÷ 2 = 78 m². Step 2: ⅔ of 78 = 78 ÷ 3 × 2 = 52 m².
🏛️ Sector Gamma — Transfer Missions
Missions 9–12 · 12 XP each · DOK 3
9
Compare & Contrast
+12 XP
📍 Sector Gamma, Site 9: Two shapes share the same base and height — but their areas tell a different story.
Parallelogram: base = 10 cm, height = 8 cm. A triangle has the same base and height. How many triangles are needed to equal the parallelogram's area?
triangles
📜 Field Notes — Show Your Work
Parallelogram: 10 × 8 = 80 cm². Triangle: (10 × 8) ÷ 2 = 40 cm². 80 ÷ 40 = ?
📓 Field Log (TWR)
The triangle formula includes ÷ 2 because it equals exactly the parallelogram's area. So triangles always equal one parallelogram.
10
Error Analysis
+12 XP
📍 Sector Gamma, Site 10: A field assistant made a calculation error on this ruin marker. Find the mistake.
Marcus says: "Triangle with base 8 and height 5 has area 40 sq units — I just did 8 × 5." Find Marcus's error and give the correct area.
sq units
📜 Field Notes — Show Your Work
Marcus used the parallelogram formula (b × h), not the triangle formula. A = (b × h) ÷ 2 = (8 × 5) ÷ 2.
📓 Field Log (TWR)
Marcus's mistake was . The correct formula is . The answer is 20 because .
11
Transfer
+12 XP
📍 Sector Gamma, Site 11: Two ruin chambers have equal floor area — one rectangular, one rhombus-shaped.
A rhombus and rectangle share the same area. Rectangle: 9 cm × 8 cm. Rhombus: diagonal d₁ = 12 cm. Find diagonal d₂.
cm
📜 Field Notes — Show Your Work
Rectangle area = 72 cm². Set rhombus = 72: (12 × d₂) ÷ 2 = 72 → 12 × d₂ = 144 → d₂ = 12.
12
Open-Ended · Cost
+12 XP
📍 Sector Gamma, Site 12: Final mission — calculate the cost to tile the expedition's triangular command center floor.
Triangle floor: base = 15 ft, height = 8 ft. Tiles = 1 ft² squares at $3.50 each. Minimum total cost? (You can't cut tiles.)
dollars
📜 Field Notes — Show Your Work
Step 1: Area = (15 × 8) ÷ 2 = 60 ft². Step 2: Need 60 tiles (can't cut). Step 3: 60 × $3.50 = $210.
⚒️ Design Lab — Parallelogram (A = 60 ft²)
There are infinite valid answers. Find any base and height that multiply to 60.
—
⚒️ Design Lab — Triangle (A = 60 ft²)
For a triangle, (b × h) ÷ 2 = 60, so b × h = 120.
—
🏛️ Composite Ruin Challenge
This ancient building cross-section combines a parallelogram (base=140, h=80) and a triangle (base=140, h=60), all in feet. Find the total cross-section area.
ft²
✍️ Expedition Journal — Prove Your Thinking
📓 Trial I — Explain the Connection
The triangle formula is half of the parallelogram formula because . I can prove this by .
📓 Trial II — Apply Decomposition
When a complex shape is decomposed into simpler shapes, the total area equals because .
📓 Trial III — Compare Formulas
A rhombus has two formulas: A = b × h and A = (d₁ × d₂) ÷ 2. I use A = b × h when I know , and A = (d₁ × d₂) ÷ 2 when I know . Both give the same result because .
🗺️ Expedition Waypoints
Base Camp
Vault
Missions
Gauntlet
Summit
⚖️ Explorer Self-Assessment
How well can you find areas of these shapes?
1
Still learning. The formulas are new territory.
2
Can do it with hints or by referencing the scrolls.
3
Can solve most missions independently.
4
Can teach this and apply it to new expeditions.
📓 Reflection Log
My mastery level is _____ because . One concept I still want to explore is .
⭐ Gate I — Identify
A parallelogram has base = 11 cm and height = 6 cm. Which formula applies, and what is the area?
⭐⭐ Gate II — Explain
A triangle and a parallelogram share the same base and height. Explain why the triangle's area is exactly half the parallelogram's area. Use a TWR sentence.
TWR Frame
The triangle formula includes ÷ 2 because , which is similar to .
⭐⭐⭐ Gate III — Transfer
A rhombus has an area of 84 in². One diagonal measures 14 in. Find the other diagonal. Then explain: could a triangle with the same diagonal lengths have the same area? Why or why not?
Saves your answers, field notes, and XP to PDF
EduWonderLab — Area Quest
Grade 6 · CCSS 6.G.A.1 · Interactive Digital Activity · Neft.Alba · For classroom use only. © EduWonderLab