Before you begin
A room asks two different questions. How much baseboard runs along the bottom of its walls? That is the walk around the room, and it is a length, measured in feet. How much carpet covers its floor? That is the fill, and it is measured in square feet, the square units of Quiz 26. The first is perimeter. The second is area. They are answered with different arithmetic, they come out in different units, and the most common error on this page is answering one when the other was asked.
Perimeter: the walk around
To find a perimeter, walk the edge and add every side. A rectangle has two lengths and two widths, so its perimeter is length plus width, doubled: 2 × (12 + 8) = 40 feet. A square has four equal sides, so its perimeter is four times one side. Any shape with straight sides works the same way, and there is no formula to remember beyond add the sides.
Area: the fill
Area counts the unit squares that cover the inside. Quiz 4 built multiplication out of exactly this picture: a rectangle 23 tiles across and 14 down holds 23 × 14 = 322 tiles. A garden bed 12 feet by 8 feet holds 12 × 8 = 96 squares a foot on each side, and that is what 96 square feet means. Written short, it is 96 sq ft, or 96 ft², with the small 2 saying the unit has been squared, as km² did in Quiz 26. A square is a rectangle whose sides match, so its area is one side times itself.
Same fence, different gardens
Perimeter and area are not tied to each other. Forty feet of fence can enclose 75 square feet or 100, depending on the shape, and among rectangles the square always holds the most. So knowing one of the two tells you nothing certain about the other, and a question that gives you the fence and asks for the floor has to give you the shape as well.
A room that is not a rectangle
A floor plan with a corner missing is two rectangles pushed together, so cut it along a line that makes rectangles, find each area, and add: 80 + 36 = 116 square feet. The other road gives the same number: the whole 14 by 10 rectangle is 140 square feet, and the missing corner is 6 by 4 = 24, so 140 − 24 = 116. Two roads, one floor.
The perimeter is still the walk around. Going clockwise from the bottom left: 10 up, 8 across, 4 down into the notch, 6 across, 6 down, and 14 back along the bottom: 10 + 8 + 4 + 6 + 6 + 14 = 48 feet. Notice that it equals 2 × (14 + 10), the perimeter of the full rectangle. Cutting a corner out of a rectangle moves two of its sides inward but does not shorten the walk, and that is worth knowing on a test: the walk around an L is the walk around the box it fits in.
Half a rectangle
A triangle with a right angle in one corner is exactly half of a rectangle, cut along the diagonal, and every triangle can be cut and rearranged into one. So the area of a triangle is half of base times height: ½ × b × h. That is Quiz 11's of at work, one half of the rectangle. A triangular garden bed with a 6-foot base and a height of 4 feet holds ½ × 6 × 4 = 12 square feet. Its perimeter is still the sum of its three sides, and the height is not one of them unless the triangle happens to have a square corner there.
Double the sides, four times the floor
Quiz 25 drew it with squares and Quiz 26 did it with a decimal, and here it becomes a tool. Double every side of a rug and its perimeter doubles, because a perimeter is made of sides. Its area goes up four times, because area is a side times a side, and both were doubled: 2 × 2 = 4. Triple the sides and the area is nine times as much. A 4 by 6 rug is 24 square feet; an 8 by 12 rug is 96, and 96 is four times 24, not twice.
The formulas on the GED sheet
| Shape | Perimeter | Area |
|---|---|---|
| Rectangle | 2 × (length + width) | length × width |
| Square | 4 × side | side × side |
| Triangle | add the three sides | ½ × base × height |
The test prints these on its formula sheet, along with formulas for circles and other shapes that get their own day. What the sheet cannot do is tell you which one the question is asking for. That is the whole skill: fence or floor, feet or square feet, before a single number is multiplied.
Three to study before you start
The garden bed in the Guide is 12 feet by 8 feet. Fencing costs $4 a foot, and a bag of soil covers 8 square feet. The fence is the perimeter: 2 × (12 + 8) = 40 feet, so 40 × $4 = $160. The soil is the area: 12 × 8 = 96 square feet, and 96 ÷ 8 = 12 bags. Two questions, two answers, two units, and the word covers in the second was the signal that it wanted the area.
Carpet is sold by the square foot, at $3. The L-shaped room in the Guide is 116 square feet, so the carpet costs 116 × $3 = $348. The baseboard along the walls is the perimeter, 48 feet, and that number never touches the carpet bill. If you had multiplied 48 by $3 you would have the price of carpeting a strip one foot wide around the walls, which nobody sells.
A corner garden bed is a triangle with a base of 6 feet, a height of 4 feet, and two sloping sides of 5 feet each. Area: ½ × 6 × 4 = 12 square feet. Perimeter: 6 + 5 + 5 = 16 feet. The 4 is used for the area and not for the perimeter, because the height runs up the middle of the bed and no fence runs there.
Now you
Work without a calculator, on paper. Questions 3 and 8 are multiple choice: choose the one best answer. Before every question, decide: fence or floor, feet or square feet.
- A rug is 9 feet by 5 feet. What is its perimeter?
- What is the area of the same rug?
- Rosa is buying baseboard, the trim that runs along the bottom of every wall of a room. Which measurement does she need?
- A) The area of the room, in square feet.
- B) The perimeter of the room, in feet.
- C) The area of the room, in feet.
- D) The perimeter of the room, in square feet.
- A square floor tile is 8 inches on each side. What is its area?
- Diego has 40 feet of fence for a rectangular garden bed. He can make it 15 feet by 5 feet or 10 feet by 10 feet. Which holds more soil, and by how many square feet?
- A room is 12 feet by 9 feet with a 5-foot by 3-foot corner cut away for a closet. What is the area of the floor that is left?
ReadingA reading rest stop, for the stubborn ones.
The okapi's measurements are on the San Diego Zoo Wildlife Alliance fact sheet. Start at the table headed Body measurements: two columns, males and females, and every number in it is a length or a weight with its source named underneath. The gallery at the end of this quiz uses those numbers.
The photograph above is not from that page. It is Plate 45 of Lankester's 1910 atlas: the specimen Major Powell-Cotton presented to the British Museum.
- What is the perimeter of the floor in question 6, closet corner and all?
- Marcus found the area of a room 9 feet by 5 feet by adding: 9 + 5 + 9 + 5 = 28 square feet. What went wrong?
- A) Nothing; 28 square feet is correct.
- B) Adding the sides walks around the room; that is the perimeter, 28 feet. The floor is 9 × 5 = 45 square feet.
- C) He should have added only once: 9 + 5 = 14 square feet.
- D) The number is right, but the unit should be feet.
- A rug is 4 feet by 6 feet. A second rug has every side doubled. The second rug's area is how many times the first rug's?
- A triangular flower bed has a base of 10 feet and a height of 6 feet. What is its area?
Check your work
Open the key: after you've finished all ten
Reading your results
| Questions | The skill they test | If they gave trouble |
|---|---|---|
| 1, 2, 4 | Rectangles and squares | Add the sides for the walk around; multiply length by width for the fill. |
| 3, 8 | Fence or floor, and the unit | Decide which was asked before touching a number, and check the unit on the way out. |
| 5, 9 | Same fence, different floors; doubling | Perimeter and area move separately. Doubling every side doubles the walk and quadruples the floor. |
| 6, 7 | Rooms that are not rectangles | Cut into rectangles and add, or take the box and subtract the missing corner. The walk around an L is the walk around its box. |
| 10 | Triangles | Half of base times height. The height is not a side. |
Eight or more right: the floor and the fence are in hand, and Quiz 28 turns from shapes to a table of fifteen measurements. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news: we've found the right ground to work. One habit for a week: before any question about a room, a garden, or a rug, say out loud whether it wants the walk around or the fill, and which unit that comes in. Then come back to these same ten.
Next in the Okapi Series: Quiz 28 — Mean, Median, Mode, and Range. Fifteen okapi skulls, measured in London in 1910, and one question their owner could not settle by looking: are there two kinds of okapi, or one? A table answers it.
Why a giraffe is not a large okapi
The okapi and the giraffe are the only two living members of their family, and the giraffe stands about 3.3 meters at the shoulder where the okapi stands about 1.5. That is 3.3 ÷ 1.5 = 2.2 times as tall. So here is a question this quiz can answer: if you took an okapi and enlarged it 2.2 times in every direction, what would you get?
Not a giraffe. Enlarge a shape and its lengths go up by the factor, its areas by the factor squared, and its volume, which is what weight follows, by the factor cubed. At 2.2 times: the hide would cover 2.2 × 2.2 = 4.84 times as much, and the animal would weigh 2.2 × 2.2 × 2.2 = 10.6 times as much. An okapi weighs about 250 kilograms, so the enlarged okapi would weigh about 2,660. The heaviest giraffe in the reference we used weighs 1,930 kilograms, well short of 2,660. The giraffe is not an okapi made bigger. It is an okapi stretched: the neck and the legs grew long, and the body did not swell to keep up. Lankester's plates show it in bone. The giraffe's neck vertebrae, on Plate 21, are not scaled-up copies of the okapi's on Plate 20; they are longer in shape.
There is a reason no animal can simply be scaled up, and it is this quiz's arithmetic. The strength of a leg bone goes with the area of its cross-section, which grows with the square. The weight the leg must carry grows with the cube. Enlarge any animal without changing its shape and its legs fall behind its body: at 2.2 times, ten and a half times the weight arrives on less than five times the bone. That is why the elephant's legs are pillars and the deer's are sticks, and why a giraffe had to become a new shape rather than a bigger okapi. Galileo worked this out in 1638 with no animals at all, from squares and cubes.
The numbers here come with their sources, which the rest stop pointed at: the okapi's shoulder height and weight from a 2013 field study, as printed on the San Diego Zoo Wildlife Alliance's fact sheet; the giraffe's shoulder height, head height and weight from Grzimek's Animal Life Encyclopedia. One other animal page we read gives the giraffe a shoulder height of four meters and more. That is the height of its head wearing the shoulder's name, and it is the kind of number this series has taught you to check rather than copy.
Prints only this section, with the pictures at full size.