Grade 7 · Grade 7: Scale, Circles and Angles · Scale, Circles and Angles
Scale Drawings and Triangle Constructions
Work with scale factors for length and area, and find out when three measurements fix a unique triangle.
Learning objectives
- Compute actual lengths and areas from scale drawings.
- Reproduce a drawing at a different scale.
- Investigate which conditions determine a unique triangle.
- Describe cross sections of prisms and pyramids.
AERO Mathematics alignment
AERO.M7.GM.1
Solve problems involving scale drawings of geometric figures, computing actual lengths and areas and reproducing a drawing at a different scale.
AERO.M7.GM.2
Draw geometric shapes with given conditions and notice the conditions that determine a unique triangle, more than one triangle or no triangle.
AERO.M7.GM.3
Describe the two-dimensional figures that result from slicing right rectangular prisms and right rectangular pyramids.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
An architect draws a room 8 cm long at a scale of 1:50 and then reprints the plan at double the size.
Think about it
How does doubling every length change the area of the drawing?
Hint: Try it with a simple 2 cm by 3 cm rectangle.
EXPLAIN
Scale, triangles and slices
A scale of 1:50 means every 1 unit on the drawing represents 50 units in reality. So 8 cm on paper is 400 cm, or 4 metres.
| Scale factor k | Length | Area | Volume |
|---|---|---|---|
| 2 | ×2 | ×4 | ×8 |
| 3 | ×3 | ×9 | ×27 |
| k | ×k | ×k² | ×k³ |
When do three measurements fix a triangle?
- SSS, SAS and ASA determine exactly one triangle.
- SSA can give two different triangles, one, or none.
- AAA fixes the shape but not the size — it gives similar triangles.
Cross sections
Slicing a right rectangular prism parallel to a face gives a rectangle; slicing at an angle can give other quadrilaterals. Slicing a rectangular pyramid parallel to the base gives a smaller rectangle.
Where this is used
Architects, map makers, 3D printing software and medical scanners all rely on scale reasoning and cross sections.
INVESTIGATION
Try it yourself
Draw a scale plan of your room at 1:50 on squared paper. Calculate the real floor area from your drawing, then redraw one wall at 1:25 and check that the area scales by four.
Watch
Introduction to Transformations
Khan Academy · 7:22
Prepares you for scale drawings and similar figures.
Before you watch: If lengths double, what happens to the area?
- A model uses a scale of 1:50. A wall is 8 cm long on the model. How long is it in reality?
- Do three given angles determine a unique triangle?
Interactive simulation · PhET
Area Model Algebra
Scale a rectangle by a factor of 2 and 3, and record what happens to the area each time.
While you explore
- When lengths double, what happens to the area?
- Why is the area scale factor the square of the length scale factor?
Key vocabulary
- Cross section
- The shape you see when you slice a solid.
- Scale factor
- The number every length is multiplied by.
Practice questions
0/1 correct
Level 1 · Criterion A
A plan uses a scale of 1:200. A line is 6 cm on the plan. What is the real length?
MYP criterion tasks
Level 2 · Criterion B
A photo is enlarged by a scale factor of 3. Describe what happens to its perimeter and area.
Reflect & track
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