Learn›Mathematics KnowledgeMK›Area & perimeter
Area and perimeter
Fence or lawn? Perimeter is the distance around a shape; area is the space inside it. Most mistakes come from mixing the two up or forgetting a ½. Here are the formulas and the habits that prevent both.
Perimeter goes around; area fills inside
Perimeter adds up the lengths of the sides, so it is measured in plain units (feet, meters). Area counts how many unit squares fit inside, so it is measured in square units (square feet, m²).
Read which one is asked
“Fence,” “border,” “trim” and “distance around” mean perimeter. “Carpet,” “paint,” “tiles,” “cover” and “surface” mean area. The test always offers the other one as a wrong answer.
How to solve them
-
1
Rectangles and squares
A rectangle has two lengths and two widths; a square has four equal sides.
RectangleP = 2l + 2wA = l × wSquareP = 4sA = s²Next step: TrianglesPerimeterA rectangle 9 m by 4 m
2 × 9 + 2 × 4 = 26 m
AreaThe same rectangle
9 × 4 = 36 m²
A squareSide 11 inches
P = 44 in, A = 121 in²
-
2
Triangles: half of base × height
A triangle is half of the rectangle drawn around it. The height is the straight-up distance from the base to the top corner, at a right angle to the base, not a slanted side.
A = ½ × base × height
Next step: Parallelograms & trapezoidsHalf of a 10 × 8 rectangle: ½ × 10 × 8 = 40. -
3
Parallelograms and trapezoids
A parallelogram is a slanted rectangle: base × height, with the height measured straight up. A trapezoid has two parallel bases of different lengths: average them, then multiply by the height.
ParallelogramA = b × hheight straight upTrapezoidA = ½(b₁ + b₂) × haverage the basesNext step: Find a sideExampleA trapezoid has bases of 9 and 15 and a height of 6. What is its area?
- Average of the bases: (9 + 15) ÷ 2 = 12.
- 12 × 6 = 72.
Answer 72 square units
-
4
Find a side from the area or the perimeter
Put what you know into the formula and solve for what is missing, exactly like an equation.
Next step: Square unitsExampleA rectangle has an area of 91 and a length of 13. Find its width and its perimeter.
- Width: 91 ÷ 13 = 7.
- Perimeter: 2 × 13 + 2 × 7 = 26 + 14 = 40.
Answer Width 7, perimeter 40
-
5
Square units convert squared
One yard is 3 feet, but one square yard is 3 × 3 = 9 square feet. Convert the lengths first, or multiply by the conversion twice.
Next step: Doubling sidesExampleA room is 4 yards by 5 yards. What is its area in square feet?
- In feet the room is 12 ft by 15 ft.
- 12 × 15 = 180 square feet.
- Same result: 4 × 5 = 20 square yards, and 20 × 9 = 180.
Answer 180 square feet
-
6
Double the sides: twice the perimeter, four times the area
Multiplying every side by a number multiplies the perimeter by that number, but the area by that number squared.
Original3 by 5 rectangle
P = 16, A = 15
Sides × 26 by 10
P = 32 (× 2), A = 60 (× 4)
Sides × 39 by 15
P = 48 (× 3), A = 135 (× 9)
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Area when perimeter is asked
A 6 by 9 rectangle, perimeter:
Wrong6 × 9 = 54
Right2 × 6 + 2 × 9 = 30
The slanted side as the height
Parallelogram: base 10, slanted side 6, height 5:
Wrong10 × 6 = 60
Right10 × 5 = 50
Forgetting the ½ for a triangle
Base 12, height 7:
Wrong12 × 7 = 84
Right½ × 12 × 7 = 42
Square units converted like plain units
2 square yards in square feet:
Wrong2 × 3 = 6
Right2 × 9 = 18
5 questions at test level
Pick an answer, then read why. No timer, no account.
A rectangular garden is 15 feet long and 8 feet wide. How many feet of fence are needed to go around it?
Show the answer
B. Fence around it is the perimeter: 2 × 15 + 2 × 8 = 30 + 16 = 46. 120 is the area, and 23 counts only two sides. It needs 46 feet.
What is the area of a triangle with a base of 14 cm and a height of 9 cm?
Show the answer
C. A = ½ × 14 × 9 = ½ × 126 = 63. 126 forgets the ½. The area is 63 square cm.
A square has a perimeter of 36 inches. What is its area?
Show the answer
B. Each side is 36 ÷ 4 = 9 inches, so the area is 9 × 9 = 81. 324 squares 18 instead of 9, and 36 repeats the perimeter. The area is 81 square inches.
A rectangle has an area of 84 square meters and a length of 12 meters. What is its perimeter?
Show the answer
C. Width: 84 ÷ 12 = 7. Perimeter: 2 × 12 + 2 × 7 = 24 + 14 = 38. 7 is only the width, and 19 adds one length and one width. The perimeter is 38 meters.
A room is 4 yards by 6 yards. How many square feet of carpet cover the floor?
Show the answer
D. In feet the room is 12 ft by 18 ft, and 12 × 18 = 216. 24 is the area in square yards, and 72 multiplies by 3 only once. It takes 216 square feet.
10 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 10 cards known.
You finished area and perimeter.
Practice score: not taken yet Take the practice
Next topic: CirclesSee this topic move.
In the course, every topic is drawn step by step on a whiteboard while a voice explains it, with timed practice tests and a plan for your test date. The first lessons of every module are free.
