Perimeter, Area and Volume

1 · Perimeter: the distance around

Imagine an ant walking all the way round the edge of a shape, starting at a corner and stopping when it gets back to where it began. The total distance it walks is the perimeter. The word comes from Greek: peri means "around" and metron means "measure". So perimeter is simply "the measure around".

Because perimeter is a distance, it is measured in ordinary length units: millimetres (mm), centimetres (cm), metres (m) or kilometres (km). A fence, a picture frame, the white line round a football pitch and the crust round a slice of toast are all perimeters.

A rectangle is a four-sided shape whose corners are all right angles (square corners, 90°). Its opposite sides are equal, so it has two long sides (the length) and two short sides (the width). Press the button to walk the ant round one side at a time. Each orange dot is 1 cm along the way.

Notice the pattern. The ant always walks length, width, length, width. So for any rectangle:

perimeter = length + width + length + width = 2 × (length + width)

2 · Area: how many squares fit

Area measures how much flat surface a shape covers: the inside, not the edge. Think of tiling a bathroom floor or covering a table with sticky notes. To measure area we need a "unit" of surface, just as we used 1 cm as a unit of length. The obvious choice is a little square that is 1 cm long and 1 cm wide. It is called one square centimetre, written 1 cm². The small ² is a reminder that two lengths (1 cm across and 1 cm up) were used to make it.

So "the area is 15 cm²" means exactly this: 15 of those little squares fit inside, with no gaps and no overlaps. Drag the purple corner (or use the sliders) to make any rectangle and count the squares.

You never actually need to count one by one. The squares come in neat rows. A 5 cm by 3 cm rectangle has 3 rows with 5 squares in each, so there are 3 × 5 = 15 squares. That is why:

area of a rectangle = length × width

A square is a rectangle with all four sides equal, so a square with side 4 cm has area 4 × 4 = 16 cm². (This is where the word "squared" for 4² comes from.)

3 · Triangles: half a rectangle

A triangle doesn't fill whole squares neatly, so counting is awkward. Instead we use a trick: draw a rectangle around it.

Two new words first. The base is the side the triangle sits on. The perpendicular height (or just "height") is how tall the triangle is, measured straight up from the base at a right angle to it. "Perpendicular" means "meeting at a right angle", like the corner of a page. The height is the dashed orange line: it is not one of the sloping sides.

The dashed line cuts the rectangle into two smaller rectangles. Each small rectangle is cut in half by a sloping side of the triangle: one half is inside the triangle (teal) and the other half is outside (plum). The numbers show each piece's area, and they match in pairs. So the triangle is exactly half of the rectangle. Drag the purple top corner left and right and watch: the area never changes.

area of a triangle = ½ × base × height

"½ ×" means "half of", so you can also multiply base × height and then divide by 2. Both give the same answer.

4 · Parallelograms: cut and slide

Two lines are parallel if they run in the same direction and never meet, like railway tracks. A parallelogram is a four-sided shape whose opposite sides are parallel. It looks like a rectangle that has been pushed over sideways.

Here is the trick. Cut off the triangle at one end along the dashed height line, then slide it across to the other end. It fits perfectly, and the parallelogram becomes a rectangle. Nothing was added or thrown away, so the area hasn't changed. Move the slider (or press Play).

area of a parallelogram = base × height

5 · Trapezia: two make a parallelogram

A trapezium (plural trapezia) is a four-sided shape with exactly one pair of parallel sides. We call the two parallel sides a and b, and the perpendicular distance between them is the height h. Think of the side view of a bucket, or a slice of a pyramid.

The trick this time: take a second, identical trapezium, turn it upside down and push it against the first. Together they make a parallelogram! Its base is a + b (one of each side joined end to end) and its height is still h. Move the slider to rotate the copy into place.

The parallelogram has area (a + b) × h, and it is made of two identical trapezia, so one trapezium is half of it:

area of a trapezium = ½ × (a + b) × h

In words: add the parallel sides, multiply by the height, then halve. The brackets matter: you add a and b first (BIDMAS).

6 · Composite shapes: split it up

A composite shape is one made by joining simpler shapes together. The L-shape below is a good example. There is no special formula for it, and you don't need one: split it into rectangles, find each area, then add. Or think of it as a big rectangle with a piece cut out, and subtract.

Choose a method. Every method gives the same answer, which is a great way to check your work.

7 · Volume: how many cubes fit

Now step up from flat shapes to solid ones. Volume measures how much space a 3-D object takes up, or how much it holds: how much water fills a tank, how much air is in a room. Just as area counts squares, volume counts cubes. A cube that is 1 cm long, 1 cm wide and 1 cm tall is called one cubic centimetre, written 1 cm³. It is about the size of a sugar cube or a dice. The small ³ shows that three lengths made it.

A cuboid is a box shape: six rectangular faces, all corners right angles. Think of a cereal box or a brick. Fill one with 1 cm cubes, one layer at a time.

The bottom layer is just a rectangle of cubes: length × width of them. The height tells you how many identical layers are stacked up. So:

volume of a cuboid = length × width × height

A cube with side 3 cm holds 3 × 3 × 3 = 27 cm³. (That is why 3³ is called "3 cubed".)

Length, area, volume: the units tell the story

8 · Prisms: cross-section × length

A prism is a solid that has the same shape all the way through, like a stick of rock, a Toblerone bar or a loaf of sliced bread. That shape is called the cross-section: it is the face you would see if you sliced straight through the prism. A cuboid is a prism whose cross-section is a rectangle.

Think of a prism as a pile of identical 1 cm thick slices. Each slice has volume equal to the area of the cross-section (a slice of area 6 cm² and thickness 1 cm holds 6 cubes). If the prism is 8 cm long there are 8 slices. So multiply!

volume of a prism = area of cross-section × length

This is the same idea as the cuboid. For a cuboid, the cross-section is a rectangle with area length × width, and its "length" through the prism is the height. So length × width × height is just a special case.

Worked examples

Quick check

What you've learned

• Perimeter is the distance all the way round the edge. Add every side. It is a length, in mm, cm, m or km. For a rectangle: 2 × (length + width).
• Area is how many 1-unit squares fit inside, in mm², cm², m². Rectangle: length × width (rows × squares in each row).
• Triangle = ½ × base × perpendicular height, because it is exactly half of the rectangle round it.
• Parallelogram = base × perpendicular height, because cutting off one end and sliding it across makes a rectangle.
• Trapezium = ½ × (a + b) × h, because two copies make a parallelogram with base a + b.
• Composite shapes: split into simple shapes and add, or take a big shape and subtract the missing part. Find missing sides by subtracting.
• Volume is how many 1-unit cubes fit inside, in cm³ or m³. Cuboid: length × width × height (one layer × number of layers).
• Prism: area of cross-section × length. Find the 2-D area first, then multiply.
• Always use the perpendicular height, convert to the same units before calculating, and check the unit: one length → cm, two → cm², three → cm³.