Scale Drawings

📐 Scale Drawings

Part 1 of 5 — What Is a Scale Drawing?

Topics in This Part

🔑 Key Concept: A scale drawing shows a real object at a different size — usually smaller — but keeps every length in the same proportion. A map, a floor plan, and a model car are all scale drawings.

Scale Drawings in Real Life

You can't fit a real house on a sheet of paper, so an architect shrinks it. A scale drawing is a picture that is bigger or smaller than the real thing while keeping the same shape — all the lengths are reduced (or enlarged) by the same amount.

Two drawings that are scale copies of each other are similar: same shape, different size.

Where you see them

💡 A scale drawing can enlarge too — a biology diagram of a cell is far bigger than the real cell. Either way, the rule is the same: every length changes by the same factor.

Reading a Scale

Every scale drawing comes with a scale that tells you how a length on the drawing matches a length in real life. It is written as a ratio:

$\text{drawing length} : \text{actual length}$

Example: A scale of $1\text{ cm} : 5\text{ m}$ means:

Every 1 cm on the drawing stands for 5 m in real life.

So a wall drawn $3$ cm long is really $3 \times 5 = 15$ m long.

⚠️ Order matters. The first number is always the drawing length and the second is the actual length. Don't swap them.

Concept Check 🎯

The Scale Factor

The scale factor is the single number you multiply a drawing length by to get the actual length (after matching the units).

For a scale of $1\text{ cm} : 5\text{ cm}$, the scale factor is $5$ — actual lengths are $5$ times the drawing lengths.

$\text{scale factor} = \frac{\text{actual length}}{\text{drawing length}}$

🔑 Watch the units. To get a pure scale factor, both lengths must be in the same unit. The scale $1\text{ cm} : 1\text{ m}$ becomes $1\text{ cm} : 100\text{ cm}$, so the scale factor is $100$ — not $1$.

Find the Scale Factor 🧮

Convert each scale to the same units, then give the scale factor (the number you multiply drawing lengths by).

1) $1\text{ cm} : 8\text{ cm}\;\Rightarrow\;$ scale factor $= \,?$ 2) $1\text{ cm} : 2\text{ m}\;\Rightarrow\;$ scale factor $= \,?$ (1 m = 100 cm) 3) $1\text{ ft} : 1\text{ yd}\;\Rightarrow\;$ scale factor $= \,?$ (1 yd = 3 ft)

Putting the Words Together

Three terms describe a scale drawing — make sure you can tell them apart:

💡 The scale keeps its units ($1$ cm to $5$ m); the scale factor is just a number once both lengths share a unit.

Match the Word 🔽

A blueprint uses the scale $1\text{ cm} : 4\text{ m}$. Pick the term that fits each blank.

What You Have So Far

You can now read a scale, know that scale drawings keep the same shape, and you can find the scale factor by matching units.

In Part 2 we use the scale to do the most common job: take a length on the drawing and find the real length.

🔑 Remember: drawing $\xrightarrow{\;\times\,\text{scale factor}\;}$ actual.

📐 Scale Drawings

Part 2 of 5 — From Drawing to Real Life

🔑 The Job: You measure a length on the drawing and need the actual length. Multiply by the scale, keeping the units lined up.

Drawing Length → Actual Length

Set up a proportion using the scale. If the scale is $1\text{ cm} : 5\text{ m}$, then:

$\frac{1\text{ cm}}{5\text{ m}} = \frac{\text{drawing}}{\text{actual}}$

Worked Example: scale $1\text{ cm} : 5\text{ m}$

A hallway is drawn $4$ cm long. How long is it really?

Each $1$ cm stands for $5$ m, so multiply the drawing length by $5$:

$4\text{ cm} \times 5\frac{\text{m}}{\text{cm}} = 20\text{ m}$

✅ Check: $4$ cm is $4$ times the $1$ cm unit, so the real length is $4$ times $5$ m $= 20$ m. ✓

Worked Example: scale $1\text{ in} : 8\text{ ft}$

A bedroom is drawn $3.5$ in wide on a floor plan. Find the real width.

$3.5\text{ in} \times 8\frac{\text{ft}}{\text{in}} = 28\text{ ft}$

Worked Example: a map, scale $1\text{ cm} : 50\text{ km}$

Two cities are $6$ cm apart on the map. The real distance is:

$6\text{ cm} \times 50\frac{\text{km}}{\text{cm}} = 300\text{ km}$

💡 Shortcut: going from drawing to actual, you always multiply by the actual-per-drawing number (here $50$). Lengths get bigger when the drawing is a shrunk-down copy.

Concept Check 🎯

Your Turn

Now do three on your own. Read the scale carefully each time — the second number changes — and remember the move is always multiply when you go from the drawing to the real world.

Drawing → Actual 🧮

Find the actual length for each. (Just type the number.)

1) Scale $1\text{ in} : 8\text{ ft}$. Drawing $= 4$ in. Actual $= \,?$ ft 2) Scale $1\text{ in} : 8\text{ ft}$. Drawing $= 6.5$ in. Actual $= \,?$ ft 3) Scale $1\text{ cm} : 50\text{ km}$. Drawing $= 7$ cm. Actual $= \,?$ km

The One Rule for This Part

Every problem so far used the same move:

$\text{actual length} = \text{drawing length} \times (\text{actual per drawing unit})$

The next check makes you say which operation you'd use before you compute — naming the move is what stops the classic mistake of dividing when you should multiply.

Pick the Right Move 🔽

For a scale of $1\text{ cm} : 20\text{ km}$, choose the correct piece of each step.

📐 Scale Drawings

Part 3 of 5 — From Real Life Back to the Drawing

🔑 The Reverse Job: You know the actual length and need to figure out how long to make it on the drawing. This time you divide by the scale.

Actual Length → Drawing Length

Drawing to actual was multiply. Going the other way, actual to drawing, is the opposite — divide.

$\text{drawing length} = \frac{\text{actual length}}{\text{scale factor}}$

Worked Example: scale $1\text{ cm} : 4\text{ m}$

A real wall is $20$ m long. How long should it be on the drawing?

Each $4$ m of real length is $1$ cm on paper, so divide by $4$:

$\frac{20\text{ m}}{4\,\text{m per cm}} = 5\text{ cm}$

✅ Check (reverse it): $5$ cm $\times 4 = 20$ m. ✓ Going forward and backward should undo each other.

Worked Example: scale $1\text{ in} : 12\text{ ft}$

A room is $60$ ft long. On the plan it should be:

$\frac{60\text{ ft}}{12\,\text{ft per in}} = 5\text{ in}$

Worked Example: a half-unit answer, scale $1\text{ cm} : 4\text{ m}$

A fence is $30$ m long. On the drawing:

$\frac{30\text{ m}}{4\,\text{m per cm}} = 7.5\text{ cm}$

💡 Drawing lengths are usually smaller numbers than the real lengths — that's the whole point of shrinking a big object onto paper.

Concept Check 🎯

Your Turn

Three to try. These all go actual $\to$ drawing, so every one is a division. Don't be surprised if an answer comes out as a decimal like $7.5$ — that's a perfectly good drawing length.

Actual → Drawing 🧮

Find the drawing length for each. (Just type the number; decimals like 7.5 are fine.)

1) Scale $1\text{ cm} : 4\text{ m}$. Actual $= 20$ m. Drawing $= \,?$ cm 2) Scale $1\text{ cm} : 4\text{ m}$. Actual $= 30$ m. Drawing $= \,?$ cm 3) Scale $1\text{ in} : 12\text{ ft}$. Actual $= 60$ ft. Drawing $= \,?$ in

Keeping the Two Directions Straight

You now have both directions. The only thing to decide each time is which one a problem is asking for:

⚠️ If your "drawing length" comes out bigger than the real length, you multiplied when you should have divided. A drawing of a building should have small numbers.

Forward or Backward? 🔽

Decide which operation each problem needs, then finish it. Scale is $1\text{ cm} : 10\text{ m}$ throughout.

📐 Scale Drawings

Part 4 of 5 — Areas and Re-Scaling

🔑 The Surprise: Lengths scale by the scale factor, but areas scale by the factor squared. We'll see exactly why, then reproduce a drawing at a brand-new scale.

Area Under a Scale

Find the actual lengths first, then compute the area from those. Don't try to scale the area directly until you've seen the pattern.

Worked Example: scale $1\text{ cm} : 3\text{ m}$

A room is drawn as a rectangle $5$ cm by $4$ cm. Find its real area.

Step 1 — actual lengths: $5\text{ cm} \times 3 = 15\text{ m}, \qquad 4\text{ cm} \times 3 = 12\text{ m}$

Step 2 — actual area: $15\text{ m} \times 12\text{ m} = 180\text{ m}^2$

The drawing's area is $5 \times 4 = 20\text{ cm}^2$. Compare:

$\frac{180}{20} = 9 = 3^2$

🔑 Big idea: The real area is the drawing area times the scale factor squared. Here the length factor is $3$, so the area factor is $3^2 = 9$.

Concept Check 🎯

The Safe Way Through an Area Problem

You can either square the scale factor, or — to be safe — just convert each side to its real length first and multiply. Both give the same answer; converting sides first is harder to mess up.

$\text{drawing sides} \xrightarrow{\;\times\,\text{scale}\;} \text{real sides} \xrightarrow{\;\times\;} \text{real area}$

🔑 Try the next drill that way: find both real side lengths, then multiply them for the area.

Areas 🧮

Scale is $1\text{ cm} : 3\text{ m}$ for all three.

1) A rectangle is drawn $6$ cm by $2$ cm. Real length of the long side $= \,?$ m 2) Same rectangle: real length of the short side $= \,?$ m 3) Real area of the rectangle $= \,?$ m²

Reproducing a Drawing at a New Scale

Sometimes you must redraw a figure at a different scale. The trick: go through the actual size.

$\text{old drawing} \xrightarrow{\;\times\,\text{old scale}\;} \text{actual} \xrightarrow{\;\div\,\text{new scale}\;} \text{new drawing}$

Worked Example

A square is drawn with $6$ cm sides at a scale of $1\text{ cm} : 2\text{ m}$. Redraw it at $1\text{ cm} : 3\text{ m}$.

Step 1 — find the actual size: $6\text{ cm} \times 2 = 12\text{ m}$.

Step 2 — apply the new scale: $\dfrac{12\text{ m}}{3} = 4\text{ cm}$.

So the new drawing has $4$ cm sides.

💡 A larger number in the scale (more meters per cm) makes a smaller drawing — you're packing more real distance into each centimeter.

Re-Scale It 🧮

A line is drawn $8$ cm long at a scale of $1\text{ cm} : 5\text{ m}$.

1) What is the actual length? $= \,?$ m 2) Redraw at $1\text{ cm} : 10\text{ m}$. New drawing length $= \,?$ cm 3) Redraw at $1\text{ cm} : 2\text{ m}$. New drawing length $= \,?$ cm

📐 Scale Drawings

Part 5 of 5 — Real-World Practice & Mastery Check

You can now read a scale, find the scale factor, go drawing $\to$ actual and back, handle areas, and re-scale a figure. Let's put it together on real problems.

Quick Reference

⚠️ Two classic traps: mixing up which way to multiply/divide, and forgetting that area uses the factor squared, not the plain factor.

Mixed Practice 🔽

A floor plan uses the scale $1\text{ in} : 6\text{ ft}$. Fill in each blank.

Maps Work Exactly the Same Way

A map is just a scale drawing of the ground. The same two moves apply:

• map distance $\to$ real distance: multiply by the scale number
• real distance $\to$ map distance: divide by the scale number

And for an area on the map, convert both sides to real distances first, then multiply. Try all three in the next challenge.

Map Challenge 🧮

A map has a scale of $1\text{ cm} : 25\text{ km}$.

1) Two towns are $8$ cm apart on the map. Real distance $= \,?$ km 2) A river is really $150$ km long. On the map it is $\,?$ cm 3) A park is drawn as a $2$ cm by $3$ cm rectangle. Its real area $= \,?$ km²

Models and Enlargements

Scale drawings aren't only maps and floor plans. A model car, train, or plane is a 3-D scale copy, written like $1 : 24$ (model : real). And a scale drawing can enlarge — a magnified diagram has a scale factor bigger than $1$ for lengths, and its area still grows by the factor squared. The last two practice questions cover both.

Mixed Practice 🎯

You're Ready

Before the exit quiz, run the whole method through your head one last time:

1. Read the scale ($\text{drawing} : \text{actual}$).
2. Drawing $\to$ actual? Multiply. Actual $\to$ drawing? Divide.
3. Area? Convert each side to real length, then multiply — areas use the factor squared.

🔑 Three questions, all using these moves. Take your time and check the direction each one asks for.

Exit Quiz ✅

Answer all three to finish the lesson.