Compound shapes and parts of shapes

Splitting and subtracting

A compound shape is made from simpler shapes whose areas you already know: rectangles, triangles, trapeziums, circles and sectors. There are two ways to find its area.

• Split it into simple pieces, find the area of each, and add.
• Draw the larger simple shape around it, find that area, and subtract the pieces that are missing.

Both give the same answer; choose whichever needs fewer, easier calculations. Before calculating, write any missing lengths on the diagram. Missing lengths come from the given ones: in an L-shape, the two horizontal pieces on the top add up to the whole width at the bottom.

Set the working out clearly, naming each piece, because each correct piece earns method marks even if a later step goes wrong.

Split along the dashed line into rectangles P and Q, or subtract the missing corner from the 12 by 9 rectangle.

Perimeters of compound shapes

The perimeter is the distance round the outside only. Lines you drew to split the shape are inside it and are not part of the perimeter.

A good habit is to start at one corner and go round the shape, writing down every edge until you get back to the start. That way no edge is missed or counted twice.

When part of the edge is curved, use arc lengths: a semicircle on a side of length $d$ adds $\frac{1}{2}\pi d$, and the straight side it replaces is no longer part of the perimeter. In terms of $\pi$, a perimeter often has a $\pi$ part and a whole-number part, such as $(32+4\pi )$ cm.

An interesting fact: the L-shape above has the same perimeter as the full 12 by 9 rectangle, $2\times (12+9)=42$ cm, because cutting out a corner just moves two edges inwards.

The dashed line is inside the window, so it is not part of the perimeter.

Parts of shapes

Many questions ask for a part of a shape: a shaded region, a ring, the part of a square outside a quarter circle. Almost always the method is subtraction: whole minus what is not wanted.

• A ring (annulus) between circles of radius $R$ and $r$ with the same centre: area $=\pi R^2-\pi r^2$. Do not work out $\pi (R-r{)}^2$; that is a different, smaller number.
• A square with a quarter circle drawn from one corner: shaded region $=$ square $-$ quarter circle.
• A semicircle, quarter circle or sector: use the fraction of the whole circle, as in the circles topic.

For the perimeter of a part, trace its boundary. It may include arcs, parts of sides and radii; list each one.

Compound solids

A compound solid is made by joining simple solids: a cone on top of a cylinder (a pencil, a rocket), a hemisphere on a cylinder (a grain silo), a cone on a hemisphere (an ice cream).

Volume: add the volumes of the parts, or subtract a part that has been removed (a hole drilled through a block, a hollow in a mould).

Surface area: this needs more care. Add the areas that are on the outside only. Where two solids are joined, the joining faces are hidden inside, so they are left out. For a cone on a cylinder of the same radius, the surface is the cone's curved surface, the cylinder's curved surface and the one circular base, and not the circle where they meet.

Read the question for what counts: "the outside surface", "excluding the base", "an open container". When a hollow is cut into a solid, its inner surface becomes part of the surface area.

Parts of solids

A part of a solid is usually a half or another simple fraction.

• Hemisphere (half a sphere) of radius $r$: volume $\frac{1}{2}\times \frac{4}{3}\pi r^3=\frac{2}{3}\pi r^3$; curved surface $\frac{1}{2}\times 4\pi r^2=2\pi r^2$. A solid hemisphere also has a flat circular face of area $\pi r^2$, so its total surface area is $3\pi r^2$. A bowl has only the curved surface.
• Half a cylinder (a trough) cut along its length: half the volume, half the curved surface, plus two semicircular ends and, if closed, a flat rectangle across the top.
• A cylinder or prism filled only partly, to a given depth, holds the volume of the solid with that depth as its height.