Area of rectangles

Area is the amount of flat region a plane figure covers, measured in square units such as $\text{cm}^2$ or $\text{m}^2$. It answers "how much surface," while perimeter answers "how far around" — a different question entirely, and one of the most common places students mix the two up. For a rectangle with length $l$ and breadth $b$:

where: $l$ is the length and $b$ is the breadth, both measured in the same linear unit, giving $A$ in that unit squared. A square is simply a rectangle whose length and breadth are equal, so with side $s$:

A rectangle's perimeter, $P = 2(l + b)$, measures the boundary length. It is worth stating clearly: two rectangles can share the same perimeter and still have very different areas, and two rectangles can share the same area and still have very different perimeters. Area and perimeter simply do not move together. Rectangles are also the reference shape for the whole chapter. A parallelogram can be cut and rearranged into a rectangle of the same base and height; that is exactly how its area formula gets derived later. Composite figures — an L-shaped room, a T-shaped plot — are measured by splitting the irregular outline into two or more rectangles, finding each area with $A = l \times b$, and adding the results. This "split into known shapes, then add" strategy is first met here and reused constantly for the rest of the chapter. One more property matters: a rectangle's area does not change when it is merely rotated or moved. Area belongs to the region itself, not to its orientation on the page — an assumption that becomes essential once shapes start getting cut and reassembled to prove other formulas.

A rectangular classroom floor measures $8\ \text{m}$ by $6\ \text{m}$. New tiles cost ₹250 per square metre. Find the area of the floor and the total cost of tiling it.

Step 1. Identify the formula needed. The floor is a rectangle with length $l = 8\ \text{m}$ and breadth $b = 6\ \text{m}$, so its area comes from $A = l \times b$.

Step 2. Substitute the values and compute the area.

Step 3. Multiply the area by the cost per square metre to get the total tiling cost.

The floor has an area of $48\ \text{m}^2$, and tiling it costs ₹12,000.

Competitive stretch

A square garden has a perimeter of $60\ \text{m}$. A square flower bed inside it has a side equal to one-third the garden's side. Find both areas.

Step 1. A square's perimeter is $P = 4s$, so $4s = 60 \implies s = 15\ \text{m}$.

Step 2. The garden's area is $A_{\text{garden}} = s^2 = 15^2 = 225\ \text{m}^2$.

Step 3. The flower bed's side is $s_{\text{bed}} = 15/3 = 5\ \text{m}$, giving $A_{\text{bed}} = 5^2 = 25\ \text{m}^2$. The garden covers $225\ \text{m}^2$, and the flower bed inside it covers $25\ \text{m}^2$ — just over a ninth of the garden, since halving a square's side quarters its area and a third of the side gives a ninth of the area.

• Confusing area and perimeter: doubling every side of a rectangle doubles its perimeter but quadruples its area, since both length and breadth double in the product.
• Mixing units before multiplying: if length is in metres and breadth is in centimetres, convert both to the same unit first — multiplying mismatched units gives a meaningless number.
• Writing area with a linear unit instead of a square one — "Area $= 48\ \text{m}$" is wrong; it must be $48\ \text{m}^2$, because area comes from multiplying two lengths together.
• Assuming equal perimeters mean equal areas (or the reverse): a $1 \times 9$ rectangle and a $3 \times 3$ square both have area $9$, but perimeters $20$ and $12$ — the two quantities simply do not track each other.

Ask the tutor: Still unsure why length times breadth gives area, or want more practice splitting composite figures into rectangles? Ask the AI tutor to walk through extra examples with you.