Build It With Area: Triangles, Parallelograms & Beyond
You are joining the Riverside Community Garden design team. Every plot, path, and shade-sail on the map is a polygon — and your job is to measure the space using area. Tú eres parte del equipo de diseño del jardín comunitario.
1 Engage
Hook & essential question
The garden team needs to cover a triangular herb bed with mulch. One bag of mulch covers 9 square feet. Looking at the bed below, how many bags should they buy?
Herb bed (drawn to scale). The dashed blue line is the height.
Quick think (no calculator): Make a first guess. Is it closer to 2 bags, 4 bags, or 8 bags? Write your guess and one reason in your notebook. You will check it in the Apply section.
Essential question: How can I break a real space into shapes I know, so I can measure exactly how much material it needs?
2 Explore
Investigate before the lesson
Explore each tool below for a few minutes. As you go, watch for one big idea: every area formula comes from rearranging a shape into a rectangle.
Cut-and-slide experiment
Drag your eyes across these two figures. The parallelogram on the left has a triangle "cut" from one end and "slid" to the other — turning it into the rectangle on the right. The base and height never change, so the area is the same.
Parallelogram
Same shape, rearranged → rectangle
Notice: Both have area = base × height = 10 × 6 = 60 square units. That is why a parallelogram uses A = b × h
3 Explain
The math, in plain language
Area is the number of unit squares that fit inside a flat shape. We always measure it in square units (like ft², m², or square units). Here are the four formulas for this unit, each connected to a rectangle.
- base (b)
- a side you measure along · la base
- height (h)
- straight distance, square to the base · la altura
- area (A)
- space inside, in square units · el área
- compose / decompose
- join shapes / break a shape apart · componer / descomponer
Parallelogram
A = b × h
Slide the triangle to make a rectangle. Multiply base × height.
Triangle
A = ½ × b × h
Two identical triangles make a parallelogram, so a triangle is half.
Trapezoid
A = ½ × (b₁ + b₂) × h
Average the two parallel bases, then multiply by the height.
Composite figure
A = A₁ + A₂ + …
Cut it into simple shapes, find each area, then add (or subtract a hole).
Worked example — triangular herb bed
Back to the Engage bed: base = 12 ft, height = 6 ft.
A = ½ × b × h = ½ × 12 × 6 = ½ × 72 = 36 ft²
One bag covers 9 ft², so bags = 36 ÷ 9 = 4 bags. (How close was your Engage guess?)
Level 1 · support Step-by-step helper / Ayuda paso a paso
- Find the base and the height. The height is always the dashed square line, not a slanted side.
- Write the formula first, then put the numbers in.
- For a triangle, do b × h first, then take half (divide by 2).
- End your answer with square units (ft², m², units²). Siempre escribe unidades cuadradas.
Level 2 · enrichment Push your thinking
The "height" can fall outside a stretched (obtuse) triangle. Why does A = ½bh still work? Sketch an obtuse triangle, drop the height, and explain how the cut-and-slide idea still rearranges it into the same rectangle.
4 Apply
Show what you can do — auto-checked
Level 1 · support Sentence starters for explaining
"To find the area I used the formula ____. I multiplied ____ × ____, then I ____. My answer is ____ square ____."
"Para hallar el área usé la fórmula ____. Multipliqué ____ × ____ y luego ____. Mi respuesta es ____ unidades cuadradas."
Level 2 · enrichment Reverse it
A triangular bed has area 48 ft² and a height of 8 ft. Work backwards to find the base. Then write a one-sentence rule for finding a missing dimension when you already know the area.
Teacher Notes & Answer Key (not printed)
Stage-by-Stage Notes (Engage → Explore → Explain → Apply → Reflect)
- Engage: estimate how much turf a shaped garden bed needs.
- Explore: decompose polygons into triangles/rectangles interactively.
- Explain: derive the triangle, parallelogram, and trapezoid area formulas.
- Apply: answer key below.
- Reflect: students explain decomposing composite figures.
Apply — Answer Key
- Q1 — Triangle (b 12, h 6): ½ × 12 × 6 = 36 ft².
- Q2: 4.
- Q3 (MC): answer b.
- Q4 — Composite total: 35 ft².
- Q5 — Composite total: 54 ft².
Standard
CCSS 6.G.A.1.
5 Reflect
Think about your thinking
A. Every area formula in this unit connects back to one shape. Which shape, and why is that connection useful?
B. Where in real life (sports field, room, screen, garden) would you need to decompose a figure to find its area? Describe it.
C. One thing that is now clear to me / Una cosa que ahora entiendo bien:
Your reflections save with this HyperDoc when you use Save as PDF or Save as DOC above.
6 Extend
Optional challenge project
Design a garden plot for the Riverside team
On grid paper, design a garden made of at least three different polygons (include one triangle and one trapezoid). Then:
- Label every base and height.
- Find the area of each plot and the total.
- If seed costs $2 per square foot, find the total seed cost.
Diseña un jardín con al menos tres polígonos y calcula el área total y el costo.