🟩Area

Reflooring a room, how many square meters of wood do you buy? Framing a photo, how long a strip do you cut? These are two different questions: one asks for the area — how much ground a shape covers; the other asks for the perimeter — how far around it is. Today belongs to area.

Area is counting squares

The idea of area is wonderfully plain: tile the shape with unit squares. The patch of ground covered by a little square with 1 cm sides is 1 square centimeter, written $1\text{ cm}^2$. Whatever number of such squares fills the shape — that is its area.

RectangleTriangle

Width w5Height h4

Area = length × width=5 × 4 = 20unit squares

Rectangles: why length times width

Tile a 6-by-4 rectangle: each row holds 6 squares, and there are 4 rows. Multiplication does the counting:

$6 \times 4 = 24$

24 little squares, so 24 square centimeters. Length × width is nothing to memorize — it is "how many per row × how many rows", the same counting you do for theater seats or floor tiles.

The units multiply too: cm × cm = square centimeter (cm²). That word "square" is exactly this multiplying — a square with 3 cm sides has area $3 \times 3 = 9$ square centimeters, not 6.

Triangles: half a rectangle

A triangle has area base × height ÷ 2. Why half? Because two identical copies of a triangle snap together into a rectangle (a general triangle first makes a parallelogram, and one straight cut turns a parallelogram into a rectangle). The rectangle covers base × height, so the triangle covers half of it:

$A = \frac{1}{2} \times b \times h$

Here b is the base and h is the height perpendicular to it. Careful: the height is the straight drop from the tip to the base line, not the slanted side — the most common slip in triangle problems.

Same fence, different field

Perimeter is the fence; area is the grass inside. Equal fences can enclose very different fields. Rectangles with perimeter 16:

Width × height	Area
1 × 7	7
2 × 6	12
4 × 4	16

From 7 to 16 — more than double. The pattern: with a fixed perimeter, the closer to a square, the bigger the area. Circles join the same family with $\pi r^2$; they are waiting in Circles.

Check yourself

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Quick quiz

↺ Try again

1. 1. A rectangle is 7 cm by 4 cm. What is its area?

   11 square cm22 square cm28 square cm

2. 2. A triangle has base 10 cm and height 6 cm. What is its area?

   60 square cm30 square cm16 square cm

3. 3. Two rectangular gardens both have perimeter 16 m. Which has the bigger area?

   The 2 m × 6 m oneThe 4 m × 4 m oneThey are equal