Welcome to the World of Area and Volume!

Hello everyone! In this chapter we build your S1 toolkit for measuring flat shapes and simple solids. By the end you'll be able to find the area of all the common 2D shapes, estimate the area of wiggly irregular shapes, find the volume of cubes and cuboids, and convert between metric units without breaking a sweat. (Arcs, sectors and cylinders are waiting for you in S2, and pyramids, cones and spheres in S3 — one step at a time!)

Part 1: Areas of Basic 2D Shapes

Area measures the amount of surface inside a shape, in square units such as \(\text{cm}^2\) or \(\text{m}^2\). You already know two old friends:

  • Rectangle: \(\text{Area} = \text{length} \times \text{width}\)
  • Square: \(\text{Area} = \text{side} \times \text{side} = \text{side}^2\)

Triangle

A triangle is exactly half of a rectangle sliced along its diagonal, so:

\(\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}\)

Watch out: the height must be the perpendicular height — the straight-up distance from the base to the top vertex, not a slanted side!

Parallelogram

Imagine slicing a triangle off one end of a parallelogram and sliding it to the other end — you get a rectangle with the same base and height. So:

\(\text{Area of parallelogram} = \text{base} \times \text{height}\)

Again, "height" means the perpendicular distance between the two parallel sides, not the slanted side.

Trapezium

A trapezium has one pair of parallel sides (call them \(a\) and \(b\)). Its area is the average of the two parallel sides times the height:

\(\text{Area of trapezium} = \frac{1}{2}(a + b) \times h\)

Memory aid: "half the sum of the parallel sides, times the height".

Rhombus

A rhombus (a "pushed-over square" with 4 equal sides) has a lovely shortcut using its two diagonals \(d_1\) and \(d_2\):

\(\text{Area of rhombus} = \frac{1}{2} \times d_1 \times d_2\)

This works because the diagonals cut the rhombus into 4 right-angled triangles that fit into half the surrounding rectangle.

Key Takeaway for Part 1

Every formula here comes from the rectangle! Triangle = half a rectangle; parallelogram = a rearranged rectangle; trapezium = average width × height; rhombus = half of the diagonal rectangle. If you forget a formula in a test, redraw the shape inside a rectangle and rebuild it.


Part 2: Composite and Irregular Shapes

Composite Figures: Split or Subtract

A composite figure is made of basic shapes stuck together. Two strategies:

  • Split: Cut the figure into rectangles, triangles and trapeziums, find each area, and add.
  • Subtract: Draw a big simple shape around the figure, then subtract the parts that don't belong (great for L-shapes and frames).

Example: An L-shaped floor can be split into two rectangles — or seen as a big rectangle with a corner bite subtracted. Both give the same answer; pick whichever needs fewer measurements.

Estimating Irregular Areas with a Grid

Leaves, lakes and hand prints don't have formulas! To estimate their area, lay a square grid (say, 1 cm squares) over the shape and count:

  1. Count every square that is completely inside the shape.
  2. Count the squares that are partly inside, and take roughly half of them (or pair them up).
  3. \(\text{Estimated area} \approx \text{full squares} + \frac{1}{2} \times \text{partial squares}\), in \(\text{cm}^2\).

A finer grid (smaller squares) gives a better estimate — that's a big idea you'll meet again much later in maths!

Key Takeaway for Part 2

Complicated flat shapes are handled by splitting, subtracting, or grid-counting. No new formulas needed — just strategy.


Part 3: Into 3D — Volume of Cubes and Cuboids

Volume measures the space a solid takes up, in cubic units such as \(\text{cm}^3\) or \(\text{m}^3\). Picture filling a box with 1 cm sugar cubes: the number of cubes that fit is the volume in \(\text{cm}^3\).

  • Cuboid (rectangular box): \(V = \text{length} \times \text{width} \times \text{height}\)
  • Cube: all edges equal, so \(V = \text{edge}^3\)

Example: A pencil box is 20 cm long, 8 cm wide and 5 cm tall. \(V = 20 \times 8 \times 5 = 800\ \text{cm}^3\).

Composite solids made of stacked cuboids work just like composite areas: split them into boxes and add the volumes.

Key Takeaway for Part 3

Volume of a box = length × width × height. Always check all three measurements are in the same unit before multiplying!


Part 4: Capacity and Unit Conversions

Capacity is the volume of liquid a container can hold. The link between capacity units and volume units is beautifully simple:

  • \(1\ \text{mL} = 1\ \text{cm}^3\) (a sugar-cube of water is exactly 1 mL!)
  • \(1\ \text{L} = 1000\ \text{mL} = 1000\ \text{cm}^3\)

Converting Area and Volume Units — the Classic Trap

1 m = 100 cm, but 1 \(\text{m}^2\) is NOT 100 \(\text{cm}^2\)! Because area is two-dimensional and volume is three-dimensional, the conversion factor gets squared or cubed:

  • \(1\ \text{m}^2 = 100 \times 100 = 10\,000\ \text{cm}^2\)
  • \(1\ \text{m}^3 = 100 \times 100 \times 100 = 1\,000\,000\ \text{cm}^3\)

Example: A fish tank has volume \(0.024\ \text{m}^3\). In litres: \(0.024 \times 1\,000\,000 = 24\,000\ \text{cm}^3 = 24\ \text{L}\).

Key Takeaway for Part 4

1 mL = 1 \(\text{cm}^3\), 1 L = 1000 \(\text{cm}^3\), and when converting units of area or volume, remember to square or cube the length factor.


Part 5: Putting It All Together

Example: A garden is a trapezium with parallel sides 8 m and 12 m, and height 5 m. A square pond of side 2 m sits inside it. Find the area of the grass.

  1. Trapezium area: \(\frac{1}{2}(8 + 12) \times 5 = \frac{1}{2} \times 20 \times 5 = 50\ \text{m}^2\)
  2. Pond area: \(2^2 = 4\ \text{m}^2\)
  3. Grass area: \(50 - 4 = 46\ \text{m}^2\)

Common Mistakes to Avoid

  • Using a slanted side as the height of a triangle, parallelogram or trapezium — height is always perpendicular.
  • Mixing units — convert everything to the same unit before calculating.
  • Writing area answers in cm or volume answers in \(\text{cm}^2\) — the unit must match the dimension: \(\text{cm}^2\) for area, \(\text{cm}^3\) for volume.
Chapter Summary
  • Triangle \(\frac{1}{2}bh\) · Parallelogram \(bh\) · Trapezium \(\frac{1}{2}(a+b)h\) · Rhombus \(\frac{1}{2}d_1 d_2\)
  • Composite shapes: split or subtract. Irregular shapes: estimate with a grid.
  • Cuboid \(V = lwh\), cube \(V = a^3\).
  • \(1\ \text{mL} = 1\ \text{cm}^3\), \(1\ \text{L} = 1000\ \text{cm}^3\), \(1\ \text{m}^2 = 10\,000\ \text{cm}^2\), \(1\ \text{m}^3 = 1\,000\,000\ \text{cm}^3\).

Next stop, S2: circles, arcs, sectors, prisms and cylinders. See you there!