5.5 Compound Shapes and Parts of Shapes


2026 Syllabus Objectives

By the end of these notes, you will be able to:

  1. Carry out calculations and solve problems involving the perimeters and areas of compound shapes and parts of shapes.
  2. Carry out calculations and solve problems involving the surface areas and volumes of compound solids and parts of solids.
  3. Give answers in terms of π where required (for example, finding the volume of half a sphere).
  4. Work with special cases like a frustum (a cone with its top cut off) and find its surface area and volume.

Part 1: Compound Shapes and Parts of Shapes (2D)


What Is a Compound Shape?

A compound shape (sometimes called a composite shape) is a flat (2D) shape that is made up of two or more simpler shapes joined together. For example, an L-shape is really just two rectangles put together.

A part of a shape is exactly what it sounds like — a portion or fraction of a standard shape. For example, a semicircle is half of a full circle.

The key idea for both is simple: break the shape into pieces you already know how to deal with, work out what you need for each piece, then combine your answers.


Key Formulas to Remember (2D)

You need these formulas as building blocks. Make sure you know them before trying compound problems.

ShapeAreaPerimeter/Circumference
Rectanglelength × width2(length + width)
Triangle½ × base × heightsum of all three sides
Circleπr²2πr
Semicircle½πr²πr + 2r
Quarter circle (quadrant)¼πr²½πr + 2r
Trapezium½(a + b)hsum of all four sides
Parallelogrambase × heightsum of all four sides

r = radius of a circle (the distance from the centre to the edge) π (pi) ≈ 3.14159… — a special number used in all circle calculations. Sometimes your answer must be left in terms of π, meaning you write π as a symbol rather than using its decimal value.


Perimeter of a Compound Shape

The perimeter is the total distance all the way around the outside edge of a shape.

Steps:

  1. Draw or study the shape carefully.
  2. Identify every edge that forms the outer boundary.
  3. Calculate the length of each outer edge separately.
  4. Add all the outer edge lengths together.

⚠️ Watch out: When two shapes are joined together, the edges where they meet are no longer on the outside. Do not include those internal edges in your perimeter.

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