The globe opens up地球儀裂開攤平
Watch the globe split along the back and spread into a flat sheet. Notice the North Pole: on the globe it is one point, and on the map it becomes the whole top edge.
看地球儀從背面裂開，攤成一張平面。注意北極：在地球儀上它是一個點，到了地圖上卻變成最上面的一整條邊。
Geography · Unit 1 · Lesson 1 · 地理 單元一 第一課
Why Is Every Flat Map Wrong?為什麼每一張平面地圖都是「錯」的？
Peel a 3D globe open and flatten it five different ways. Watch equal circles swell and squash, drag Greenland down to the equator, and find out why no flat map can get everything right.
把 3D 地球儀剝開，用五種方法攤平。看一樣大的圓怎麼被拉大、壓扁，把格陵蘭拖到赤道，再想想為什麼沒有一張平面地圖能樣樣都對。
Big idea · 一句話看懂
The surface of a ball cannot be laid flat without stretching it. So every flat map has to choose what to keep right: shape, area, distance, or direction. There is no best map, only a map that fits the job.
球的表面不可能不拉扯就攤平。所以每一張平面地圖都要選：形狀、面積、距離、方向，要保住哪一樣。沒有「最好的地圖」，只有「適合這個用途的地圖」。
3D Model · 3D 模型
A Globe You Can Peel Flat 一顆可以剝開攤平的地球儀
Press “Peel it flat,” or pull the slider, and the globe opens along the back and spreads out into a map. Then choose another way of flattening it. The orange circles are all the same size on the globe, so wherever a circle grows or turns into an oval, the map has stretched the Earth. Click any spot to read how much. The model draws only land, sea, and grid lines.
按「攤平」或拉動滑桿，地球儀會從背面裂開，攤成一張地圖。接著換別種攤法看看。橘色的圓在地球儀上都一樣大，所以哪裡的圓變大了、變成橢圓了，就是地圖在那裡把地球拉扯了。點任何一個地方，可以讀出拉了多少。模型只畫陸地、海洋和經緯線。
Drag to turn the globe · 拖曳轉動地球 Click a spot to measure it · 點一下量那個地方 Scroll or pinch to zoom · 滾輪／雙指縮放
This 3D model needs WebGL, which this browser does not support. The numbers on the right still work, and so do the reading, the dragging map, and the game below.
這個瀏覽器不支援 WebGL，無法顯示 3D 模型；右邊的數字照樣能用，下面的課文、拖著走的地圖和小遊戲也都能用。
- Land (outline only; no borders are drawn)
陸地（只有外形，不畫國界） - Circles that are all the same size on the globe
在地球儀上一樣大的圓 - The spot you picked, and its line of latitude
你選的地方，和它所在的緯線 - The equator
赤道 - Taiwan
台灣
Each orange circle is 4.5° across from center to edge, about 500 km on the real Earth. All five maps are drawn for a globe of the same size, so you can compare them directly.
每個橘色的圓從圓心到邊緣是 4.5 度，在真正的地球上大約 500 公里。五種地圖都是用同樣大小的地球儀畫的，可以直接互相比較。
Land outlines: Natural Earth, 1:50m Land, version 5.1.2, public domain (naturalearthdata.com), simplified for this page. No borders or country names are shown. The stretch numbers are computed by the page itself. · 陸地外形：Natural Earth 1:50m Land 5.1.2 版，公有領域（naturalearthdata.com），經本頁簡化。不畫國界、不標國名。變形的倍率由頁面自己計算。
Reading · 英文閱讀
1
Unit 1: Why Is Every Flat Map Wrong?
Peel a tangerine in one piece, then try to press the peel flat on the table. It will not lie flat. It bulges, and if you push harder, it tears. Now think about the Earth. The Earth is round like the tangerine, and a map is flat like the table. Every flat map of the world has to solve the tangerine-peel problem somehow.
把一顆橘子的皮完整剝下來，再試著把它壓平在桌上。它不會乖乖躺平：中間鼓起來，再用力壓就裂開。現在想想地球。地球像橘子一樣是圓的，地圖像桌面一樣是平的。每一張平面的世界地圖，都得想辦法解決這個「橘子皮問題」。
A globe does not have this problem. A globe is a small model of the Earth with the same round shape, so the shapes of the lands, their sizes, the distances between places, and the directions from one place to another can all be right at the same time. But a globe is hard to carry, hard to print in a book, and it never shows you the whole world at once. So people flatten the globe. A map projection is a set of rules for moving every point on the round Earth to a point on a flat sheet.
地球儀沒有這個問題。地球儀是一顆小小的地球模型，形狀同樣是圓的，所以陸地的形狀、大小、兩地之間的距離、從一地到另一地的方向，可以同時都是對的。但地球儀不好攜帶、沒辦法印在書裡，也沒辦法一眼看到整個世界。所以人們把地球儀攤平。地圖投影就是一套規則，把圓圓的地球上的每一個點，搬到一張平面上的某一個點。
Here is the trouble: something has to stretch. In the 1820s the mathematician Carl Friedrich Gauss proved a theorem which shows that the surface of a ball cannot be flattened without changing some of its distances. Mapmakers call this stretching distortion. A flat map can keep shapes right, or keep areas right, or keep some distances or directions right, but it can never keep all of them right everywhere.
麻煩在這裡：一定有地方要被拉扯。1820 年代，數學家高斯證明了一個定理，從這個定理可以知道：球的表面不可能攤平而不改變上面的某些距離。製圖的人把這種拉扯叫做變形。一張平面地圖可以把形狀畫對，或把面積畫對，或把某些距離、某些方向畫對，但不可能在每個地方把這些全部畫對。
The most famous choice was made in 1569 by Gerardus Mercator, who drew a map for sailors. On his map, a straight line always follows one compass direction, so a sailor could draw the route with a ruler. To make that work, the map stretches more and more toward the poles. At 60 degrees of latitude everything is drawn twice as wide and twice as tall, which means four times too large in area. That is why Greenland looks about as big as Africa on a Mercator map. In fact Africa is about fourteen times larger.
最有名的一個選擇，是 1569 年麥卡托做的。他畫了一張給航海的人用的地圖：在這張圖上，一條直線永遠朝著同一個羅盤方向，所以水手可以用尺畫出航線。為了做到這一點，地圖越靠近兩極就拉得越開。在緯度 60 度的地方，每樣東西都被畫成兩倍寬、兩倍高，也就是面積大了四倍。這就是為什麼在麥卡托地圖上，格陵蘭看起來和非洲差不多大。實際上，非洲大約是格陵蘭的十四倍。
Other maps make the opposite choice. An equal-area projection keeps every size right, so Greenland shrinks back to one-fourteenth of Africa. The price is shape: on one well-known equal-area map, lands near the equator look tall and thin. Some people say equal-area maps are fairer for classrooms, because the Mercator map makes places far from the equator look larger than they are. Others answer that the Mercator map was built for steering ships and was never meant for comparing sizes. Each side is describing what its map does well.
另一些地圖做了相反的選擇。等積投影把每個地方的大小都畫對，所以格陵蘭縮回非洲的十四分之一。代價是形狀：在一種常見的等積地圖上，赤道附近的陸地看起來又高又瘦。有人說等積地圖用在教室裡比較公平，因為麥卡托地圖把離赤道遠的地方畫得比實際大；也有人回答，麥卡托地圖本來就是為了讓船照著方向走而設計的，從來不是拿來比大小的。兩邊說的，都是自己那張地圖擅長的事。
So which map is the best? It depends on the job. To hold a compass course at sea, use a Mercator map. To compare the sizes of lands, use an equal-area map. To study the Arctic, use a map with the North Pole in the middle. To see everything right at once, pick up a globe. Every flat map is wrong somewhere, and a good map reader asks two questions: what was this map made for, and what did it give up?
那麼哪一張地圖最好？要看你拿它做什麼。在海上要照著羅盤方向走，用麥卡托地圖。要比較陸地的大小，用等積地圖。要研究北極，用北極在正中央的地圖。想要樣樣都對，就拿起地球儀。每一張平面地圖都有某個地方是「錯」的；會看地圖的人會問兩個問題：這張地圖是為了什麼做的？它犧牲了什麼？
Read aloud · 課文朗讀
Questions · 閱讀理解
Why can a globe show shapes, sizes, distances, and directions correctly all at once, when a flat map cannot?
A globe has the same round shape as the Earth, so nothing needs to be stretched. A flat map must flatten a round surface, and that cannot be done without distortion, so it has to give up at least one of the four.
On a Mercator map, Greenland looks about as big as Africa. What is really going on?
The Mercator map stretches everything more and more toward the poles. Greenland is far north, so it is drawn many times too large, while Africa lies across the equator and is drawn close to its true size. In reality Africa is about fourteen times larger than Greenland.
Your class wants a wall map to compare the sizes of the continents. A ship's officer wants a chart for steering by compass. Should they buy the same map? Why or why not?
No. The class needs an equal-area map, because it keeps sizes right even though it bends shapes. The officer needs a Mercator chart, because a straight line on it keeps one compass direction even though sizes are wrong. Each map fits a different job.
Words & Phrases · 生字及片語
globe(n.)地球儀Spin the globe and find the island of Taiwan.轉動地球儀，找出台灣這座島。map projection(n.)地圖投影A map projection turns the round Earth into a flat map.地圖投影把圓的地球變成一張平面地圖。distortion(n.)變形The distortion on this map is greatest near the poles.這張地圖在兩極附近變形最大。equator(n.)赤道The equator divides the Earth into a northern half and a southern half.赤道把地球分成北半邊和南半邊。latitude(n.)緯度Taipei is at about 25 degrees north latitude.台北大約在北緯 25 度。area(n.)面積The area of Africa is about fourteen times that of Greenland.非洲的面積大約是格陵蘭的十四倍。stretch(v.)拉長；拉開This map stretches the lands that are far from the equator.這張地圖把離赤道遠的陸地拉大了。navigate(v.)航行；導航Sailors used the stars and a compass to navigate across the ocean.水手靠星星和羅盤在海上航行。
Quick Check · 小測驗
1. Why is every flat map of the world distorted somewhere?
2. What was the Mercator map of 1569 designed for?
3. At 60 degrees of latitude, a Mercator map draws areas about how many times too large?
4. What does an equal-area projection keep right?
5. Which sentence is true?
Try It in the Model · 在模型裡試試
Four things to watch 四件值得看的事
Then try it yourself. Click on Greenland, then on the middle of Africa, and compare the area numbers on each of the five maps.
然後自己動手試試。先點格陵蘭，再點非洲的中間，在五種地圖上比一比面積的倍率。
Circles that grow越來越大的圓
On the Mercator map every circle stays round, so shapes are kept. But the circles swell toward the poles. At 60° the panel reads × 2 each way and × 4 in area.
在麥卡托地圖上，每個圓都還是圓的，形狀保住了；但越靠近兩極，圓就越大。到了 60 度，右邊欄顯示兩個方向都是 2 倍，面積是 4 倍。
Circles that squash被壓扁的圓
Switch to an equal-area map. Now every circle covers the same area, but the circles turn into ovals: tall near the equator, wide near the poles.
換成等積地圖。現在每個圓蓋住的面積都一樣，但圓變成了橢圓：赤道附近是直的，兩極附近是橫的。
The orange-peel map橘子皮地圖
Cut the globe into twelve strips instead. Shapes and sizes look good, but the map is full of gaps. To avoid tearing, a map has to stretch.
改成把地球儀切成十二瓣。形狀和大小看起來都不錯，但地圖上到處是裂縫。地圖想要不裂開，就得拉扯。
Try It · 動手試試
Drag Greenland to the Equator 把格陵蘭拖到赤道
This is a Mercator map. Drag the orange shape of Greenland down to the equator and watch it shrink to the size it would have if it were drawn with the same ruler as Africa. Then drag Africa north and watch it grow. The shapes are moved on the round Earth first and then redrawn, so their real size never changes.
這是一張麥卡托地圖。把橘色的格陵蘭拖到赤道，看它縮成和非洲「用同一把尺畫」的大小。再把非洲往北拖，看它變大。這些陸塊是先在圓的地球上搬動、再重新畫出來的，所以它們真正的大小從頭到尾沒有變。
Drag a colored shape · 拖動有顏色的陸塊
Greenland 格陵蘭
- Real area真正的面積
- Where its center is now中心現在的緯度
- This map draws its area這張地圖把它的面積畫成
- How big it looks next to Africa看起來是非洲的
- How big it really is next to Africa實際上是非洲的
At home, far from the equator, this map draws the shape many times too large. Drag it toward the dashed line.在原來的位置、離赤道很遠的地方，這張地圖把它畫得比實際大了好多倍。把它往虛線拖拖看。 Watch the number in gold. The closer the shape gets to the equator, the closer the number gets to 1.注意金色的數字。陸塊越靠近赤道，數字就越接近 1。 On the equator the map draws the shape at very nearly its true size. Now the two percentages almost match.在赤道上，地圖幾乎把它畫成真正的大小。現在兩個百分比幾乎一樣了。
Game · 小遊戲
Which Map Fits the Job? 哪一張地圖適合？
Eight jobs. For each one, choose the kind of map that fits it best. There is no timer, and you can keep choosing until you find it. The game counts the jobs you match on the first try.
八個用途。每一題選出最適合的那一種地圖。不計時，可以一直選到對為止；遊戲會記下你第一次就選對的題數。
Question 1 of 8 · 第 1 題（共 8 題）
Right on the first try · 第一次就答對 0 / 0
Map English · 地圖英語
Eight sentences for talking about maps 談地圖時用得到的八句英文
Every lesson in this series has a Map English card: eight sentences for saying where things are and what a map shows. Listen, repeat, then try them with a map or globe in front of you.
這個系列每一課都有一張「地圖英語」卡：八句用英文說位置、說地圖的句子。先聽、再跟著唸，然後對著一張地圖或一顆地球儀自己說說看。
The equator runs through the middle of the map.
赤道從地圖的正中間穿過。
runs through: a line crossing a place一條線穿過某個地方
Greenland is near the North Pole.
格陵蘭在北極附近。
near: close to靠近
Taiwan is off the coast of East Asia.
台灣在東亞的外海。
off the coast of: in the sea, not far from the shore在外海、離岸不遠
Africa lies on both sides of the equator.
非洲橫跨赤道的兩邊。
on both sides of: north and south of a line在一條線的兩邊
Greenland looks bigger on this map than it really is.
格陵蘭在這張地圖上看起來比實際大。
looks … than it really is: for distortion說地圖的變形
North is at the top of the map.
地圖的上方是北邊。
at the top of / at the bottom of在上方／在下方
Australia is in the Southern Hemisphere.
澳洲在南半球。
in the Northern / Southern Hemisphere在北半球／南半球
This map keeps the sizes right, but it stretches the shapes.
這張地圖把大小畫對了，但把形狀拉長了。
keeps … but stretches …: what a map does well and badly說一張地圖保住什麼、犧牲什麼
Stories Behind the Maps · 地圖背後的故事
A sailor's map, a long argument, the map in your phone, and Taiwan's own grid 水手的地圖、一場爭論、手機裡的地圖，和台灣自己的坐標
Four short stories about people choosing what a map should keep.
四個小故事，講的都是人怎麼選擇一張地圖要保住什麼。
1569 · For Sailors · 給水手的地圖
Mercator's straight lines麥卡托的直線
Gerardus Mercator presented his world map in 1569. Part of its long Latin title means “for the use of navigators.” A course that keeps one compass direction is called a rhumb line, and on his map every rhumb line is straight. Sea charts still use the projection today. Its weakness for wall maps was known from the start: the poles cannot be shown at all, and lands far from the equator look far too large.
麥卡托在 1569 年發表他的世界地圖，那一長串拉丁文標題裡有一段的意思是「供航海者使用」。羅盤方向固定不變的航線叫做恆向線，在他的地圖上，每一條恆向線都是直的。今天的海圖仍然使用這種投影。拿來當掛圖的缺點也從一開始就看得出來：兩極根本畫不出來，離赤道遠的陸地看起來大得太多。
A Long Argument · 一場爭論
Which map belongs on the classroom wall?教室牆上該掛哪一張地圖？
In the 1970s Arno Peters promoted an equal-area map and criticized the Mercator map for making lands far from the equator look too large. In 2017 the public schools of Boston began using a map of this kind. Many mapmakers replied that the Mercator map was designed for navigation, and that the Gall–Peters map bends shapes badly. In 1989 and 1990, seven geographic organizations in North America advised against using any rectangular map as a general world map, and they named both of these. The argument shows that a map is a choice.
1970 年代，彼得斯推廣一種等積地圖，並批評麥卡托地圖把離赤道遠的陸地畫得太大。2017 年，美國波士頓的公立學校開始使用這一類地圖。許多製圖的人則回應：麥卡托地圖是為航海設計的，而高爾－彼得斯地圖把形狀扭曲得很厲害。1989 到 1990 年，北美有七個地理學會與製圖學會建議，一般用途的世界地圖不要用任何長方形的投影，這兩種都點名在內。這場爭論讓人看見：地圖是一種選擇。
In Your Pocket · 口袋裡的地圖
Why online maps are square網路地圖為什麼是正方形的
Most online maps use a form of the Mercator projection, often called Web Mercator. Google's documentation explains that the map is cut off at about 85 degrees north and south, which makes the whole world a square. A square splits neatly into four smaller squares, again and again, and those are the tiles your phone loads as you zoom in. The projection is also good at keeping angles. Zoomed all the way out, it has the same size problem as any Mercator map.
大多數的網路地圖用的是麥卡托投影的一種變化，常被稱為 Web Mercator。Google 的技術文件說明：地圖在南北緯大約 85 度的地方切掉，這樣整個世界剛好是一個正方形。正方形可以整齊地一分為四、再一分為四，這些就是你放大地圖時手機載入的一塊塊圖磚。這種投影也擅長保住角度。把畫面縮到最小時，它和所有的麥卡托地圖一樣，有大小不準的問題。
Taiwan · 台灣
A Mercator map turned on its side一張橫過來的麥卡托地圖
Taiwan's official coordinate system, TWD97, uses a projection called Transverse Mercator. Imagine the Mercator map turned on its side, so that the line of least distortion runs north–south instead of along the equator. Taiwan is a long island running north to south, so this fits it well. According to the National Land Surveying and Mapping Center, the central line for the main island is 121 degrees east, and the zone is only two degrees wide, which keeps the distortion very small.
台灣的官方坐標系統 TWD97，用的是「橫麥卡托投影」。想像把麥卡托地圖橫過來，讓變形最小的那條線不再是赤道，而是一條南北向的線。台灣是一座南北狹長的島，這種投影很適合。依內政部國土測繪中心的說明，台灣本島的中央子午線是東經 121 度，而且一帶只有經度兩度寬（二度分帶），所以變形非常小。
In the Curriculum · 對應課綱
Where this lesson fits 這一課對應的課綱
These entries come from Taiwan's 12-Year Basic Education curriculum for social studies (2018). The elementary-school content has no entry about map projections, so the links below are to junior and senior high school geography.
下面的條目出自十二年國民基本教育課程綱要社會領域（民國 107 年發布）。國小階段的學習內容沒有談地圖投影的條目，所以這裡列的是國中與高中地理。
地 Aa-Ⅳ-1社會領域・地理（國中） · Social studies, geography (junior high)
全球經緯度座標系統。
Latitude and longitude are the grid that every projection moves onto the flat map. Lesson 2 of this series is about that grid.經緯度就是每一種投影搬到平面上的那套格線。本系列第二課會專門談它。
地 Ac-Ⅴ-1社會領域・地理（普通型高中必修） · Social studies, geography (senior high, required)
地圖本質與要素。
The curriculum adds a note to this group of entries: “若介紹地圖投影時，應著重於介紹地圖投影的變形及合宜應用的案例。” (If map projections are introduced, the focus should be on their distortion and on cases of fitting use.) That is the idea of this lesson.課綱在這一組條目的說明欄寫：「若介紹地圖投影時，應著重於介紹地圖投影的變形及合宜應用的案例。」這正是本課的主題。
地 Ac-Ⅴ-2社會領域・地理（普通型高中必修） · Social studies, geography (senior high, required)
地圖種類及其用途。
The game “Which Map Fits the Job?” practices matching a kind of map to a use.「哪一張地圖適合？」這個小遊戲，練習的就是把地圖的種類和用途配起來。
Myth vs. Fact · 常見迷思
Four things people get wrong 四個常見的誤會
✗ Myth · 迷思The map on the classroom wall shows the true sizes of the lands.教室牆上的地圖，畫的就是陸地真正的大小。
✓ Fact · 事實Only if it is an equal-area map. Many wall maps and almost all online maps enlarge the lands that are far from the equator.只有等積地圖才是。很多掛圖和幾乎所有的網路地圖，都把離赤道遠的陸地放大了。
✗ Myth · 迷思The Mercator map is a wrong map, and an equal-area map is the right one.麥卡托地圖是錯的地圖，等積地圖才是對的。
✓ Fact · 事實Each keeps something different. The Mercator map keeps shapes and compass directions. An equal-area map keeps sizes. Each is right for its job and wrong for the other one.它們保住的東西不一樣。麥卡托地圖保住形狀和羅盤方向，等積地圖保住大小。各自在自己的用途上是對的，換到對方的用途上就不對。
✗ Myth · 迷思With a good enough computer, someone will make a flat map with no distortion.只要電腦夠厲害，總有一天能做出完全沒有變形的平面地圖。
✓ Fact · 事實This is a matter of geometry, and computing power cannot change it. It was proved about two hundred years ago that a sphere cannot be flattened without distortion. A map can only choose where to put it.這是幾何的問題，電腦再厲害也改變不了。大約兩百年前就有人證明：球面不可能毫無變形地攤平。地圖只能選擇把變形放在哪裡。
✗ Myth · 迷思On a map, the shortest way between two places is always a straight line.在地圖上，兩地之間最短的路一定是一條直線。
✓ Fact · 事實On the round Earth, the shortest path follows a great circle. On a Mercator map that path usually looks curved, which is why long flights seem to bend toward the poles.在圓的地球上，最短的路沿著大圓走。這條路畫在麥卡托地圖上通常是彎的，所以長途航線看起來會往兩極的方向彎。
Remember It · 記憶口訣
Maps in a sentence 一句話記住地圖
Round will not go flat圓的攤不平
Remember the tangerine peel. To lie flat, it has to stretch or tear.
想想橘子皮：要攤平，不是拉扯就是裂開。
Shape, area, distance, direction: pick形狀、面積、距離、方向：選
A flat map can keep some of the four, never all of them.
一張平面地圖可以保住其中幾樣，不可能四樣全保。
Mercator for steering, equal-area for sizes要走方向用麥卡托，要比大小用等積
One keeps compass directions straight. The other keeps sizes fair.
一個讓羅盤方向是直線，一個讓大小公平。
Ask what the map is for先問地圖是做什麼用的
There is no best map, only a map that fits the job.
沒有最好的地圖，只有適合這個用途的地圖。
Before the Activities · 做活動之前
Working safely 安全提醒
- Peel the fruit with your fingers. If the peel needs a starting cut, an adult makes it. Students do not use knives.水果用手剝。如果需要先劃一刀才好剝，由大人來劃；學生不拿刀。
- Use round-tipped scissors, cut away from your other hand, and pass scissors with the handles toward the other person.用圓頭剪刀，剪的時候刀口不要朝向另一隻手；遞剪刀時把握柄朝向對方。
- Blow up balloons only as far as the teacher shows you, and keep them away from your face when you cut them. Check first whether anyone in the class is allergic to latex.氣球只吹到老師示範的大小，剪的時候離臉遠一點。活動前先確認班上有沒有人對乳膠過敏。
- Wash your hands after handling the fruit, and wipe up juice so that nobody slips.碰過水果之後洗手，滴在地上的果汁擦乾，免得有人滑倒。
Classroom Activity 1 · 課堂活動一
The Tangerine-Peel Map 橘子皮地圖
You need · 準備材料
- One tangerine or small orange for each pair每兩人一顆橘子或小柳丁
- A marker that writes on the peel一支能畫在果皮上的筆
- A sheet of paper and a heavy book一張紙和一本厚書
- A pencil一枝鉛筆
Steps · 步驟
- Before peeling, draw a line around the middle of the fruit for the equator. Draw four or five circles of the same size on the peel, some near the equator and some near the top.剝皮之前，先在水果的腰上畫一圈當赤道。再在果皮上畫四、五個一樣大的圓，有的靠近赤道，有的靠近頂端。
- Peel the fruit with your fingers, keeping the peel in as few pieces as you can.用手把皮剝下來，盡量讓皮保持完整、越少片越好。
- Lay the peel on the paper with the outside facing up, and press it flat with the book. Watch where it tears.把皮攤在紙上、外面朝上，用書壓平。看看它從哪裡裂開。
- Trace the outline of the flattened peel on the paper. Mark the gaps.在紙上描出壓平後果皮的輪廓，把裂縫標出來。
- Look at your circles. Which ones were torn or pulled out of shape? Compare with the orange-peel map in the 3D model.看看你畫的圓，哪些被撕開、被拉得變形了？和 3D 模型裡的「橘子皮地圖」比一比。
The peel tears because it cannot stretch. A printed map does not tear because the mapmaker stretches the picture instead. Ask the class: which would you rather have, gaps or stretching?
果皮會裂開，是因為它不能伸縮。印出來的地圖不會裂開，是因為製圖的人改用「拉扯圖形」來代替。問問全班：你寧願要裂縫，還是要拉扯？
Classroom Activity 2 · 課堂活動二
The Balloon Map 氣球地圖
You need · 準備材料
- One round balloon for each group每組一個圓形氣球
- A permanent marker一支油性筆
- Round-tipped scissors圓頭剪刀
- Four pushpins or pieces of tape, and a piece of cardboard四個圖釘或幾段膠帶，和一塊厚紙板
Steps · 步驟
- Blow the balloon up to the size of a grapefruit and tie it. Draw an equator and six small squares of the same size, some near the equator and some near the knot.把氣球吹到葡萄柚那麼大，打結。畫一條赤道，再畫六個一樣大的小正方形，有的靠近赤道，有的靠近打結的地方。
- Let the air out slowly. Cut the balloon open from the knot to the opposite end.慢慢把氣放掉。從打結的地方剪到另一端，把氣球剪開。
- Stretch the rubber into a rectangle on the cardboard and fix the four corners with tape or pins. An adult helps with the pins.把橡皮在厚紙板上拉成長方形，四個角用膠帶或圖釘固定。圖釘請大人幫忙。
- Measure your squares now. Which ones grew the most? Are they still square?現在量一量那些正方形。哪幾個變大最多？還是正方形嗎？
- Open the 3D model and choose the Mercator map. Do the orange circles change in the same places as your squares?打開 3D 模型，選麥卡托地圖。橘色的圓變化的地方，和你的正方形一樣嗎？
Rubber can stretch, so the balloon map has no gaps, but the squares near the edge are pulled much bigger than the ones in the middle. That is distortion you can measure with a ruler.
橡皮可以伸縮，所以氣球地圖沒有裂縫；但靠邊的正方形被拉得比中間的大很多。這就是可以用尺量出來的變形。
Go Further · 延伸閱讀
The round Earth in other lessons, and what comes next 其他課裡的圓形地球，以及下一課
Astronomy · Lesson 3 · 天文教育・第三課Why Do We Have Seasons?為什麼有四季？The same round, tilted Earth, seen from space. That lesson shows why sunlight falls differently at different latitudes.同一顆圓圓的、歪著轉的地球，從太空看。那一課說明為什麼不同緯度照到的陽光不一樣。Go to the lesson · 前往這一課 →How Things Work · Lesson 7 · 萬物原理・第七課How Does GPS Know Where You Are?GPS 怎麼知道你在哪裡？Satellites find your position on the round Earth. A map projection then decides where that position lands on your flat screen.衛星先在圓的地球上找出你的位置，地圖投影再決定這個位置要畫在平面螢幕的哪裡。Go to the lesson · 前往這一課 →Geography · Lesson 2 · 地理・第二課How Do You Find Any Place on Earth?怎麼找到地球上任何一個地方？Latitude and longitude: two numbers that work like an address for any spot on the globe.經度和緯度：兩個數字，就是地球上任何一個地方的地址。Go to the lesson · 前往這一課 →
Sources · 資料出處
Names, dates, numbers, and quotations on this page were checked against these sources (October 2026). The stretch factors and ratios are computed by the page itself and have automated tests · 本頁的人名、年代、數字與引文依下列資料查證（2026 年 10 月）；變形的倍率與比值由頁面自己計算，並有自動測試
- Land outlines: Natural Earth, 1:50m Physical Vectors, Land, version 5.1.2. Public domain. Simplified for this page; no borders or names.陸地外形：Natural Earth 1:50m Land 5.1.2 版，公有領域；經本頁簡化，不含國界與地名Natural Earth: Terms of Use ↗
- A map projection is “a systematic representation of all or part of the surface of a round body, especially the Earth, on a plane”; this “cannot be done without distortion.” Formulas for the projections, the scale factor sec φ of the Mercator projection, and Tissot's analysis of distortion (1859, 1881).地圖投影的定義、「不可能沒有變形」、各投影的公式、麥卡托的比例因子 sec φ、底索的變形分析（1859、1881）Snyder, Map Projections: A Working Manual, USGS Professional Paper 1395 (1987) ↗
- Mercator projection: presented in 1569; conformal; rhumb lines are straight; the poles cannot be shown; the 1569 title includes “ad usum navigantium.” Plate carrée credited to Marinus of Tyre, about AD 100; Mollweide presented in 1805; Gall's projection, 1855.麥卡托投影：1569 年發表、正形、恆向線是直線、畫不到極點、標題有「供航海者使用」；等距圓柱投影約西元 100 年；莫爾威投影 1805 年；高爾的投影 1855 年Snyder (1987), chapters 7, 10, 12, 31 ↗
- “Any straight line on a Mercator projection map is a line of constant true bearing”; introduced in 1569 by Gerardus Mercator; areas farther from the equator appear disproportionately large.麥卡托地圖上的直線是方位固定的線；1569 年由麥卡托提出；離赤道越遠的地區看起來越大Encyclopaedia Britannica, “Mercator projection” ↗
- “All projections involve some degree of distortion.”所有的投影都有某種程度的變形Encyclopaedia Britannica, “projection (cartography)” ↗
- Gauss's Theorema Egregium (1827): the surface of a sphere cannot be unfolded onto a plane without distorting distances.高斯的「絕妙定理」（1827 年）：球面不可能攤成平面而不改變距離Wikipedia, “Theorema Egregium” ↗
- Area of Greenland: 2,166,086 km².格陵蘭的面積：2,166,086 平方公里Statistics Greenland, “Geography” ↗
- “Africa's total land area is approximately 11,724,000 square miles (30,365,000 square km).” This figure includes the islands; the Africa shape on this page is the mainland only. 30,365,000 ÷ 2,166,086 ≈ 14.0 (our calculation).非洲的面積：約 30,365,000 平方公里（含島嶼；本頁拖動的非洲只有本土）。30,365,000 ÷ 2,166,086 ≈ 14.0（本教材計算）Encyclopaedia Britannica, “Africa” (read October 2026) ↗
- Area of mainland Australia: 7,656,127 km² (7,688,287 km² with its islands).澳洲本土的面積：7,656,127 平方公里（含島嶼為 7,688,287）Geoscience Australia, “Area of Australia – States and Territories” (updated 2014) ↗
- Boston Public Schools introduced Gall–Peters maps in 2017; supporters and critics are both quoted.2017 年波士頓公立學校開始使用高爾－彼得斯地圖；報導中並列支持與批評的說法NPR, March 21, 2017 ↗
- Arno Peters devised his projection in 1967 and presented it in 1973 as the “Peters world map”; it matches the projection James Gall described in 1855.彼得斯 1967 年設計、1973 年發表「彼得斯世界地圖」；和高爾 1855 年提出的投影相同Wikipedia, “Gall–Peters projection” ↗
- In 1989 and 1990, seven North American geographic organizations adopted a resolution recommending against any rectangular projection, including Mercator and Gall–Peters, for reference maps of the world (Robinson, 1990).1989–1990 年北美七個地理與製圖團體通過決議，建議世界參考地圖不要用任何長方形投影（含麥卡托與高爾－彼得斯）Wikipedia, “Map projection” ↗
- Globes are commonly made from long, thin strips of printed paper called gores, which narrow to a point at the poles.地球儀常用印好的細長紙條（gores）貼成，紙條在兩極收成尖端Wikipedia, “Globe” ↗
- “We cut off base map imagery utilizing the Mercator projection at approximately +/- 85 degrees to make the resulting map shape square.”Google 地圖在南北緯約 85 度切掉，讓地圖成為正方形Google Maps Platform, “Map and Tile Coordinates” ↗
- EPSG:3857, “WGS 84 / Pseudo-Mercator,” used for web mapping; the cutoff latitude 85.0511° is arctan(sinh π).EPSG:3857（Web Mercator），網路地圖使用；邊界緯度 85.0511 度OpenStreetMap Wiki, “Slippy map tilenames” ↗
- Online map services use a variation of the Mercator projection, which “is very good for preserving angles.” The shortest path between two points on a sphere is a segment of a great circle.網路地圖使用麥卡托投影的變化，它擅長保住角度；球面上兩點之間最短的路是大圓的一段National Geographic Education, “Selecting a Map Projection” and “Great Circle” ↗
- TWD97: Transverse Mercator projection in two-degree zones; central meridian 121° E for Taiwan's main island; scale factor 0.9999 on the central meridian.TWD97：「採用橫麥卡托投影經差二度分帶，其中央子午線為東經121度……中央子午線尺度比為0.9999」內政部國土測繪中心〈常用測繪名詞〉 ↗
- Chinese terms: 地圖投影 (map projection), 麥卡托投影 (Mercator projection), 等積投影 (equal-area projection), 變形 (distortion), 大圓 (great circle), 恆向線 (rhumb line), 莫爾威投影 (Mollweide projection).中文名詞依國家教育研究院樂詞網（高中以下地理學名詞、測繪學名詞）國家教育研究院 樂詞網 ↗
- Curriculum entries 地 Aa-Ⅳ-1, 地 Ac-Ⅴ-1, 地 Ac-Ⅴ-2 and the note on map projections.課綱條目 地 Aa-Ⅳ-1、地 Ac-Ⅴ-1、地 Ac-Ⅴ-2 與關於地圖投影的說明十二年國民基本教育課程綱要 社會領域（教育部，民國 107 年） ↗