Perimeter is the walk around the edge. Area is the covering inside.
Find the width
A rectangular garden has an area of 84 m² and a length of 12 m. What is its perimeter?
Two steps. First the width: 84 ÷ 12 = 7 m. Then the perimeter: 2(12) + 2(7) = 38 m. A adds the length and width once instead of twice. D doubles the area, which mixes up a square measurement with a linear one — an answer in metres can never come from doubling something in square metres.
| Shape | Perimeter | Area |
|---|---|---|
| Rectangle | 2l + 2w | lw |
| Square | 4s | s² |
| Triangle | a + b + c | bh2 |
| Parallelogram | 2a + 2b | bh |
| Trapezoid | add all four sides | (a + b)h2 |
| Circle | 2πr (the circumference) | πr² |
A triangle is half a parallelogram, which is why its area is halved. A trapezoid uses the average of its two parallel sides, a + b2, times the height.
Straight up from the base, at a right angle — not along a slanted side.
Here the perimeter needs the 7, and the area needs the 6. Using the 7 for area gives 70 instead of 60. Problems print both numbers on purpose.
r = 7 cm
use π ≈ 3.14
C = 2π(7) = 14π ≈ 43.96 cm
a distance
A = π(7)² = 49π ≈ 153.86 cm²
a surface
If a problem gives the diameter, halve it first. A circle "20 cm across" has r = 10, and using 20 makes the area four times too big.
A triangle has area 54 cm² and base 12 cm
find the height
54 = 12h2 = 6h
substitute
h = 9 cm
divide by 6
A square has area 121 m²
find its perimeter
s = √121 = 11 m
undo the squaring
P = 4(11) = 44 m
then the perimeter
A semicircle's perimeter is half the circumference plus the diameter. A ring's area is the big circle minus the small one.
A window is a semicircle with a diameter of 16 cm. Find its perimeter, using π ≈ 3.14.
curved edge = 3.14 × 16 ÷ 2 = 25.12 cm
half of πd
perimeter = 25.12 + 16 = 41.12 cm
add the straight edge
Stopping at 25.12 cm measures only the curve. The perimeter goes all the way round the shape, and a semicircle is closed by its diameter, so the 16 cm must be added: 41.12 cm.
A circular fishpond has a radius of 4 m. A walkway 3 m wide runs all the way round it. Find the area of the walkway, using π ≈ 3.14.
outer radius = 4 + 3 = 7 m
add the width once, not twice
3.14 × 7² − 3.14 × 4² = 3.14 × (49 − 16)
big circle minus the pond
= 3.14 × 33 = 103.62 m²
the walkway alone
Squaring the 3 m width, 3.14 × 9, gives a circle 3 m in radius — not a ring at all.
A rectangle has a perimeter of 34 cm and a width of 5 cm. Find its area.
l + w = 34 ÷ 2 = 17
half the perimeter is one length plus one width
l = 17 − 5 = 12 cm
the missing side
area = 12 × 5 = 60 cm²
check: 2(12) + 2(5) = 34 ✓
34 − 5 = 29 forgets that there are two widths; the perimeter has to be halved first.
A 10 cm by 7 cm card has a 3 cm by 2 cm rectangle cut from one corner. Find the area and the perimeter of what is left.
area = 10 × 7 − 3 × 2 = 70 − 6 = 64 cm²
whole minus the hole
perimeter = 2(10) + 2(7) = 34 cm
the two new edges replace the two removed ones
Cutting a corner lowers the area but leaves this perimeter unchanged: the 3 cm and 2 cm edges of the notch are just the missing pieces moved inward.
The same formulas, entered from an unusual side.
A circle has a circumference of 18.84 cm. Find its area, using π ≈ 3.14.
2 × 3.14 × r = 18.84, so r = 3 cm
18.84 ÷ 6.28 = 3
A = 3.14 × 3² = 28.26 cm²
now the usual formula
Every circle question goes through r; find it first, whatever you are given.
An obtuse triangle has a base of 6 cm, and its height, drawn outside the triangle to the extended base, is 5 cm. Find its area.
A = 6 × 52
the height may fall outside; it is still perpendicular to the base line
= 15 cm²
same formula
Only the base and the perpendicular distance to the opposite corner matter, wherever that perpendicular lands.
Find the area of a semicircle of radius 5 m, leaving the answer in terms of π.
π × 5² ÷ 2 = 25π ÷ 2
half a full circle
= 12.5π m²
no 3.14 needed
"In terms of π" means keep π as a letter; the answer is exact, not rounded.
For a hole, subtract. For an L-shape, cut it into two rectangles and add. Either way, work out each piece completely before combining.
Measurement items give a figure with extra numbers printed on purpose — a slanted side, a diameter instead of a radius — or ask for a composite shape such as a path, a ring or a cut-out. The fastest route is to choose the formula first and then pick the numbers it needs; the 10-second check is the units, cm for a perimeter and cm² for an area.
Find the area
A trapezoid has parallel sides of 8 cm and 14 cm and a height of 6 cm. What is its area?
A = (a + b)h2 = (8 + 14)62 = 1322 = 66 cm². Another way to see it: the average of the parallel sides is 11, and 11 × 6 = 66. B forgets to halve, and D is the perimeter of two sides rather than an area — notice it is in the wrong order of size entirely.
Hint: average the two parallel sides first, then multiply by the height. That is the same formula with less to get wrong.
