Restoration Series · Quiz 37 · Same Shape, Different Size

The Guide

Before you begin

The drawing is a triangle with a smaller copy of itself in the corner — the whole idea of this quiz, and the reason the trapezoid turns up at the end of it.

This is the last part of geometry the test asks about, and it is the one that shows up outside a classroom most often. Every map has a scale. Every photograph enlarged or shrunk keeps its shape. Every scale model, every architect’s drawing, every pattern graded from a size 8 to a size 16 is the same piece of mathematics.

It is also almost entirely a proportion problem, which means it is family 5's work wearing a geometry hat. If Quizzes 23 to 25 went well, this one will too.

What similar means

Two figures are similar when one is a scaled copy of the other: same shape, different size.

Said precisely, two things have to be true at once. The angles have to match — every corner of one equals the corner in the matching position on the other. And the sides have to be in the same ratio — every side of the big one is the same number of times the matching side of the small one.

Both conditions, not one. This matters, and there is a question about it further down.

The word for same shape and same size is congruent. Two congruent figures are identical; you could lay one on the other and it would disappear. Similar is the looser word: congruent figures are similar, with a scale factor of 1, but similar figures need not be congruent.

The scale factor

The scale factor is the one number that connects the two figures. Divide any side of the large figure by the matching side of the small one, and that is it.

If the scale factor is 3, every single side of the big one is 3 times the matching side of the small one. Find it once and you can find anything.

Finding it is nearly always the first move. A question gives you one pair of matching sides — that pair is there to hand you the scale factor — and then asks about some other side.

Matching the sides up

The only real difficulty in this family is knowing which side goes with which. Get that wrong and the arithmetic is beside the point.

When a question is written in letters, the letters do the work for you. If triangle ABC is similar to triangle DEF, then A matches D, B matches E, and C matches F. The order they are written in is not an accident, and it is the whole instruction. Side AB matches side DE, side BC matches side EF, side CA matches side FD.

The letters are in matching order, and that order tells you which side belongs to which.

When a question gives you a picture instead of letters, match by position: the longest side of one goes with the longest side of the other, the shortest with the shortest, and the side opposite a marked angle goes with the side opposite the matching angle.

Writing the proportion

Once the sides are matched, the work is a proportion, solved by cross-multiplying exactly as in Quiz 24.

A triangle has sides 3, 4 and 5. A similar triangle has its shortest side 9. The matching pair is 3 and 9, so the scale factor is 9 ÷ 3 = 3, and the other two sides are 4 × 3 = 12 and 5 × 3 = 15.

Or set it out as a proportion and cross-multiply, which is safer when the numbers are not friendly:

3/9 = 4/x, so 3x = 36, so x = 12.

Keep the two figures on their own sides. The most common error here is writing one fraction small-over-big and the next big-over-small. Pick one arrangement — small on top in both, or big on top in both — and keep it for the whole question. A quick check afterwards: if the scale factor is more than 1 the answer should be larger than the side you started from, and if it is less than 1 the answer should be smaller. That catches a flipped proportion every time.

The shadow question

There is one question in this family the GED asks over and over, and it is worth recognizing on sight.

A man 6 feet tall casts a shadow 4 feet long. At the same moment a tree casts a shadow 30 feet long. How tall is the tree?

The same picture as Quiz 36's Eratosthenes, on the scale of an afternoon. Parallel light, two right angles, two triangles of one shape.

The man and his shadow make a right triangle. The tree and its shadow make another. Both have a right angle where the upright meets the ground, and both have the same angle where the sunlight comes in, because the sun is far enough away that its rays arrive parallel. Two matching angles is enough for triangles, so the two are similar.

So 6/4 = h/30. Cross-multiply: 4h = 180, and h = 45 feet.

Two angles are enough, for triangles

That last step deserves its own line, because it is the one shortcut in this family.

If two angles of one triangle equal two angles of another, the two triangles are similar. You do not have to check the sides at all.

The reason is Quiz 36. The three angles of a triangle add to 180, so once two of them match, the third has no choice but to match as well. All three angles equal means the same shape, and the same shape means the sides fall into ratio on their own.

This is why the shadow question works without measuring anything but shadows, and it is why surveyors have been able to measure distances they could not walk for two thousand years.

Triangles only. This shortcut belongs to triangles and to nothing else. Four-sided shapes can have all four angles equal and still not be similar, which is exactly what question 8 is about.

What happens to area

Here is the fact the test hides a question in.

If the scale factor is 3, the sides are 3 times as long — but the area is 9 times as big, not 3.

Quiz 27 found this for garden beds and Quiz 35 for the radius of a can. It is the same fact each time: area is two dimensions, so the scale factor acts twice.

So the rule is: sides scale by the factor, area by the factor squared. Scale factor 2, area 4 times. Scale factor 5, area 25 times. And since volume is three dimensions, volume scales by the factor cubed: a model at scale 2 holds 8 times as much.

The trap is simple and it catches people. Told that two similar triangles have a scale factor of 4 and the small one has an area of 5, the answer is not 20. It is 5 × 16 = 80.

A parallel cut makes a smaller copy

Now the last shape on the formula sheet that this site has not yet covered, and it arrives through the door we have just opened.

Take a triangle and cut across it with a line parallel to the base. The piece at the top is a triangle. Because the cut is parallel to the base, that small triangle has exactly the same three angles as the whole one — and by the rule two paragraphs above, it is similar to it.

A trapezoid is a triangle with a smaller similar triangle taken off the top. That is not the definition, but it is where this one comes from.

The piece left underneath is a trapezoid: a four-sided figure with one pair of parallel sides. Those two parallel sides are called the bases, written b₁ and b₂, and the perpendicular distance between them is the height.

The area of a trapezoid

The formula sheet gives it as A = ½h(b₁ + b₂), and in plain words that is: average the two parallel sides, then multiply by the height.

Averaging the two bases is what the ½ and the addition are doing together. A trapezoid with bases of 8 and 12 has the same area as a rectangle 10 wide, because 10 is the average of 8 and 12. Multiply by the height of 5 and the area is 50.

Here is why it is true, and it takes one drawing.

Two trapezoids always fit together this way, whatever their shape. Half of the parallelogram is one trapezoid.

Turn a second copy of the trapezoid upside down and push it against the first. The slanted sides match, because they are the same side, and what you have made is a parallelogram whose base is b₁ + b₂ and whose height is h. Its area is base times height. One trapezoid is half of that.

The height is the perpendicular distance, not the slanted side. A trapezoid question will often give you the length of a slanted side as well, and it is there to be picked up by mistake. The height is the straight-across distance between the two parallel sides, measured square to them. If a question gives you four numbers and the formula needs three, one of them is bait.

The guard

Four things, and the first two carry most of the points.

Match the sides before you write anything. Say out loud, or write down, which side goes with which. The letters tell you when there are letters; the positions tell you when there are not.

Keep the proportion consistent. Both smalls on top, or both bigs on top. Then check the answer against common sense: bigger figure, bigger answer.

Area squares the factor. Sides times k, area times k², volume times k³. If a question mentions area or volume and gives you a scale factor, this is what it is asking.

For a trapezoid, use the perpendicular height. And remember what the formula is saying: average the parallel sides, multiply by the height.

Worked Examples

Three to study before you start

Example 1 · Matching, then solving

Triangle ABC is similar to triangle PQR. AB = 10, BC = 14, and PQ = 25. Find QR.

First match. The letters are in order, so AB goes with PQ and BC goes with QR. The pair we are given is AB and PQ: 10 and 25.

Scale factor: 25 ÷ 10 = 2.5. So QR = 14 × 2.5 = 35.

As a proportion instead: 10/25 = 14/QR, so 10 × QR = 350, and QR = 35. Same answer, and the proportion is the safer route when the scale factor comes out awkward. The check: 25 is bigger than 10, so the answer should be bigger than 14, and it is.

Example 2 · Area, not sides

Two similar rectangles have a scale factor of 3. The smaller has an area of 12 square inches. What is the area of the larger?

Not 36. Area scales by the square of the factor: 3² = 9, so the area is 12 × 9 = 108 square inches.

You can see it without the rule if you want. Say the small rectangle is 3 by 4, which is 12. Tripling the sides gives 9 by 12, which is 108. The sides tripled; the area went up ninefold.

Example 3 · A trapezoid

A trapezoid has parallel sides of 14 cm and 6 cm, a height of 9 cm, and a slanted side of 11 cm. What is its area?

The slanted side is not needed. Average the two parallel sides: (14 + 6) ÷ 2 = 10. Multiply by the height: 10 × 9 = 90 square cm.

Or straight into the sheet’s formula: A = ½h(b₁ + b₂) = ½ × 9 × 20 = 90. Every trapezoid question on the test gives you at least one number you do not need, so decide what the formula wants before you pick any number up.

The Quiz · Ten Questions

Now you

Work on paper. A calculator is fine. Match the sides before you write a proportion, and check each answer for size before you move on. Questions 6 and 10 have two parts.

Two similar rectangles. The smaller is 4 cm by 6 cm. The larger has a shorter side of 10 cm. How long is its longer side?
Triangle ABC is similar to triangle DEF. AB = 6 and DE = 9. If BC = 8, how long is EF?
A woman 5 feet tall casts a shadow 3 feet long. At the same moment a flagpole casts a shadow 24 feet long. How tall is the flagpole?
Two similar triangles have a scale factor of 4. The smaller one has an area of 5 square cm. What is the area of the larger one?
- A) 9 square cm
- B) 20 square cm
- C) 80 square cm
- D) 625 square cm
A trapezoid has parallel sides of 8 cm and 12 cm and a height of 5 cm. What is its area?
On a map, 1 inch stands for 25 miles. (a) Two towns are 3.5 inches apart on the map. How far apart are they on the ground? (b) Two other towns are 60 miles apart. How far apart are they on the map?

Reading A reading rest stop, for the stubborn ones.

Ten minutes with the last section of this page — The Company, below the answer key — before you finish the quiz. It is not about a mathematician this time. It is about a trade: the men and women who knelt on a wooden floor hundreds of feet long and drew a ship at full size, because a scale drawing is only as true as the scaling. It will not help you with question 7. It is the best answer I know to the question of who this mathematics is actually for.

No photograph at this rest stop. The drawings in this quiz are all above; the reading stands on its own.

A triangle has sides of 3, 4 and 5. A similar triangle has a longest side of 20. How long are its other two sides?
Ramon says that all rectangles are similar to one another, because every rectangle has four right angles. Is he right? Say why or why not.
A trapezoid has parallel sides of 6 m and 10 m, a height of 4 m, and a slanted side of 5 m. What is its area?
- A) 20 square m
- B) 32 square m
- C) 40 square m
- D) 64 square m
In the figure below, a triangle with a base of 12 and a height of 8 is cut by a line parallel to its base. The small triangle above the cut has a base of 3. (a) What is the scale factor between the small triangle and the whole one? (b) What is the area of the trapezoid left below the cut?

Answer Key

Check your work

1 15 cm

The shorter sides are the matching pair: 4 and 10. The scale factor is 10 ÷ 4 = 2.5, so the longer side is 6 × 2.5 = 15. As a proportion, 4/10 = 6/x gives 4x = 60 and x = 15. The answer is larger than 6, as it must be when the scale factor is more than 1.

2 12

The letters match A to D, B to E, C to F, so AB goes with DE and BC goes with EF. The scale factor is 9 ÷ 6 = 1.5, so EF = 8 × 1.5 = 12. If you paired BC with DE instead you would have got something else, and nothing in the arithmetic would have warned you — which is why the matching comes first and on paper.

3 40 feet

The woman and her shadow make one right triangle, the flagpole and its shadow another, and the sun’s rays arrive parallel so the two triangles have the same angles. Then 5/3 = h/24, so 3h = 120 and h = 40. Keeping heights on top in both fractions is what keeps this straight; 3/5 = 24/h would have worked too, since it puts shadows on top in both, but mixing them gives 14.4 and no warning.

4 C) 80 square cm

Area scales by the square of the factor: 4² = 16, and 5 × 16 = 80. B) 20 is 5 × 4, the answer to a question about a side rather than an area, and it is the one most people choose. D) 625 is 5⁴, squaring the wrong thing. A) 9 is nothing but a plausible-looking number. This question exists to find out whether you know the area rule, so it is worth saying it once more: sides times 4, area times 16.

5 50 square cm

Average the parallel sides: (8 + 12) ÷ 2 = 10. Times the height: 10 × 5 = 50. Or A = ½h(b₁ + b₂) = ½ × 5 × 20 = 50. Square units, because it is an area.

6 (a) 87.5 miles (b) 2.4 inches

(a) Each inch is 25 miles, so 3.5 × 25 = 87.5 miles. (b) Going the other way, divide: 60 ÷ 25 = 2.4 inches. A map scale is a scale factor with units attached, and the two halves of this question are the two directions you can travel through it — multiply to go from map to ground, divide to come back. Deciding which one the question wants, before touching the numbers, is the whole of it.

7 12 and 16

The longest sides are the matching pair: 5 and 20, so the scale factor is 20 ÷ 5 = 4. The others are 3 × 4 = 12 and 4 × 4 = 16. Check the shape survived: 12, 16, 20 is the 3, 4, 5 triangle four times over, and 12² + 16² = 144 + 256 = 400 = 20², so it is still a right triangle. Scaling never changes the angles, and that check is the Pythagorean theorem confirming it.

8 No. Equal angles are enough for triangles, but not for anything else.

A rectangle 2 by 3 and a rectangle 2 by 10 both have four right angles, and neither is a scaled copy of the other — one is stubby, the other is a strip. For figures with four sides or more, the angles matching is only half the requirement; the sides have to be in the same ratio as well. A 2 by 3 rectangle is similar to a 4 by 6 and to a 10 by 15, because those have the same ratio, and to nothing else. Triangles are the exception, and they are the exception because their three angles have to add to 180, so matching two of them forces the third and leaves the sides no freedom.

9 B) 32 square m

½ × 4 × (6 + 10) = ½ × 4 × 16 = 32. The 5-meter slanted side is not used; it is there to be picked up by mistake. C) 40 is what you get by using that 5 as the height. D) 64 is the right shape of calculation with the ½ forgotten. A) 20 uses only one of the two bases. Four numbers, three needed, and the question is mostly about which one to leave alone.

10 (a) ¼ (b) 45 square units

(a) The bases are the matching pair: 3 ÷ 12 = ¼. (b) Everything about the small triangle is a quarter of the whole, so its height is 8 × ¼ = 2. That leaves 8 − 2 = 6 for the height of the trapezoid, whose two parallel sides are 12 and 3. So ½ × 6 × (12 + 3) = ½ × 6 × 15 = 45. The other road gives the same number: the whole triangle is ½ × 12 × 8 = 48, the small one is ½ × 3 × 2 = 3, and 48 − 3 = 45. Two routes agreeing is the best check there is, and this question was built so that both are open.

Reading your results

Eight to ten: family 7, the biggest on the test, is now finished. Go and take a practice test. Five to seven: sort your misses into two piles. If they were the proportion questions — 1, 2, 3, 7 — the fault is almost always in the matching, so redo them writing down which side goes with which before anything else. If they were 4, 5, 9 or 10, the fault is in a formula, and those you can fix by reading the Guide’s last three sections again. Under five: do questions 1, 2 and 3 until finding the scale factor is automatic, because every other question in this family starts there. Then come back for the trapezoid, which is one formula and a warning about the slanted side.

The Company · An Interlude

The floor where the ship was drawn

The previous interludes in this series have been about individuals: a shopkeeper counting burials, a mathematician who wanted a diagram on his grave, a librarian with a stick. This one is about a trade, because the mathematics on this page was for two centuries somebody’s job, and the job has almost no names attached to it.

A ship begins as a drawing. The drawing is small — a sheet on a table, a few feet across, showing a hull that will be six hundred feet long. Between that sheet and the steel there was a room called the mould loft.

The mould loft was a long open floor at the top of a shipyard building, often two or three hundred feet from end to end, floored in smooth planking and usually painted black. The floor was the drawing board. On it, the lines of the ship were laid down full size.

Not enlarged in the head. Not read off a ruler. Drawn out, at one to one, in chalk and in scribed lines, by people on their hands and knees, across a floor bigger than most houses.

Ask why and the answer is the subject of this quiz. A scale drawing is similar to the real thing: same shape, sides all multiplied by the same number. That is exactly true in mathematics. In a shipyard it is exactly true only if the drawing is exactly right, and a drawing at one inch to four feet is at a scale factor of forty-eight. Every error in the drawing comes out forty-eight times as large in the steel. A line laid down a sixteenth of an inch off — less than the width of a pencil — is three inches out on the ship. Three inches, at the joint where two plates have to meet, is a ship that does not close.

So the loftsmen redrew everything at full size and corrected it there, where an error was its own size and no more. This was called fairing the lines: running long flexible wooden battens along the drawn curve, held down by heavy iron weights called ducks, and adjusting until the batten lay sweet, with no flat spots and no sudden bends. A batten will not bend unfairly. It finds the honest curve on its own, and a loftsman’s eye and hand did the rest.

From the faired full-size lines they made moulds — thin wooden patterns for each frame and each plate, which went down to the platers and the frame benders on the shop floor as the actual shape to cut and bend to. The mould loft was the place where a number on a page became a shape you could hold.

It is worth sitting with what that work was. It was mathematics: proportion, scale, the geometry of curves in three dimensions projected onto a flat floor in three views at once. It was also physical labor, done kneeling, in cold buildings, for a wage. The people who did it were tradesmen who had served an apprenticeship, and they were not called mathematicians and did not call themselves that. Some of those buildings survive: Harland and Wolff’s old drawing offices in Belfast are a hotel now.

The trade is mostly gone. From the 1960s onward the fairing was done by computer — numerical lofting, and then the whole thing in software — and the long black floors were emptied and turned to other uses. That is not a complaint. The computer does the arithmetic better and the ships are no worse.

But there is something to keep from it, and it is the thing this whole quiz is about. Scale is not free. A drawing is similar to the thing it describes only to the accuracy of the drawing, and somebody, somewhere, has to do the work of making the scaling true. For two hundred years that somebody was on their knees on a wooden floor with a batten and a piece of chalk. The mathematics was simple. The work was not, and it was skilled, and it was paid by the hour.

Where this goes. That is family 7 complete — perimeter and area, volume and surface area, angles, and now similar figures and the trapezoid. It is the biggest family on the test and every part of it now has a lesson. Its page on this site is Family 7, with Quiz 27, Quiz 35 and Quiz 36 beside this one, and every formula named here is on The Formula Sheet. The series goes next to Quiz 38, on reading graphs.