Hello IGCSE Mathematicians! Calculating Compound Shapes

Welcome to the exciting world of Mensuration! So far, you've learned how to calculate the area, perimeter, volume, and surface area of simple shapes like rectangles, triangles, and spheres. But what happens when shapes get complicated?

In this chapter, we tackle compound shapes and parts of shapes. This is where mathematics meets the real world—think about finding the area of an 'L' shaped garden, or calculating how much paint you need for a cylindrical storage tank with a hemispherical roof.

Don't worry if these shapes look complex! The secret is simple: we break the complex shapes down into the basic shapes you already know how to calculate.


Part 1: Compound 2D Shapes (Perimeter and Area)

A compound shape (or composite shape) is a two-dimensional figure made up of two or more standard geometrical shapes joined together (like rectangles, triangles, or semicircles).

1.1 Calculating Perimeter (The 'Walking the Fence' Method)

The perimeter is the total length of the outside boundary of the shape.

🔑 Key Concept & Common Mistake to Avoid:

When calculating the perimeter of a compound shape, you must ignore any internal lines where the shapes are joined together. You are only calculating the distance around the outside edge.

Step-by-Step Process for Perimeter:

  1. Identify all external edges: Look at the shape and mentally "walk" along the perimeter.
  2. Determine missing lengths: Often, you will need to use simple addition or subtraction based on the given dimensions to find the length of any unlabeled straight sides.
  3. Calculate curved lengths (if present): If the shape includes curves (like semicircles or arcs), use the relevant circumference formula (Circumference \(C = 2\pi r\) or \(C = \pi d\)) and adjust it (e.g., half the circumference for a semicircle).
  4. Sum all external lengths: Add all the side lengths and curved lengths together.

Example Analogy: If you join two rectangular desks together, the perimeter is the distance around the combined outline, not including the edge where the desks meet.

1.2 Calculating Area (The 'Cut and Conquer' Method)

The area is the total space enclosed within the boundary of the shape.

Strategy: Decomposition

Break the compound shape into familiar, non-overlapping shapes (usually rectangles, triangles, parallelograms, or trapeziums).

Step-by-Step Process for Area:

  1. Divide: Split the compound shape into simpler standard shapes (A, B, C, etc.). Draw your dividing lines clearly.
  2. Calculate Individual Areas: Use the standard area formulas for each smaller shape (e.g., \(A = lw\) for a rectangle, \(A = \frac{1}{2}bh\) for a triangle).
  3. Add the Areas: Sum the areas of all the parts to find the total area of the compound shape.

Alternatively, sometimes it's easier to use the Subtraction Method: Imagine a large, simple shape that contains your compound shape. Calculate the area of the large shape, then subtract the area of the missing "cut-out" parts.

Quick Review: Perimeter and Area of Compound 2D Shapes
  • Perimeter: Sum of external boundaries only.
  • Area: Sum of the areas of the non-overlapping component shapes.

Part 2: Calculating Parts of Circular Shapes

The syllabus requires calculations involving parts of shapes, especially sectors and arcs of a circle (C6.3/E6.3 are prerequisites, but E6.5 extends this to complex parts).

2.1 Arc Length and Sector Area

These calculations involve finding a fraction of the circle's total circumference or area, based on the central angle \(\theta\).

Remember that the angle must be a factor of \(360^{\circ}\) in Core mathematics, but the formulas work for any angle in Extended.

Formulas (Angle \(\theta\) in degrees):

  • Arc Length, \(L\):
    \(L = \frac{\theta}{360} \times 2\pi r\)
  • Sector Area, \(A\):
    \(A = \frac{\theta}{360} \times \pi r^2\)

2.2 Finding the Area of a Segment

A segment is the area enclosed by an arc and the chord connecting the endpoints of the arc. This is a classic "Part of a Shape" calculation.

Step-by-Step Process for Segment Area:

  1. Find the Sector Area: Calculate the area of the sector defined by the angle \(\theta\). (Area of Sector).
  2. Find the Triangle Area: Calculate the area of the triangle formed by the two radii and the chord.
  3. Subtract:
    Area of Segment = Area of Sector - Area of Triangle.

Tip for the Triangle Area: Since you usually know the two sides (both radii, \(r\)) and the angle between them (\(\theta\)), the formula for the area of a non-right-angled triangle is often useful here (especially in Extended): \(A = \frac{1}{2}r^2 \sin \theta\).

Remember:

If the question asks for the answer in terms of \(\pi\), do not substitute 3.142 or use the calculator's \(\pi\) button—leave your answer as an expression multiplied by \(\pi\).


Part 3: Compound Solids (Volume and Surface Area)

A compound solid is a 3D object formed by joining two or more standard solids (like a cylinder topped with a cone).

3.1 Calculating Volume (Always Add!)

Calculating the volume of a compound solid is generally straightforward. Since volume measures the space inside, you simply calculate the volume of each component and add them together.

Step-by-Step Process for Volume:

  1. Identify Solids: Break the compound solid into standard 3D shapes (e.g., Cuboid + Pyramid, or Cylinder + Hemisphere).
  2. Calculate Individual Volumes: Use the appropriate volume formula for each part. (Formulas for prism, pyramid, cylinder, cone, and sphere volumes are provided in the exam).
  3. Sum the Volumes: Add the individual volumes together.

3.2 Calculating Surface Area (The 'Wrapping Paper' Challenge)

The surface area is the total area covering the exterior of the solid. This is the hardest calculation for compound shapes because you must account for the faces that are hidden or joined together.

🚨 The Crucial Rule (The Wrapping Paper Analogy):

Imagine you are wrapping the entire compound solid in paper. Any surface that is glued to another surface (i.e., the face where they join) is an internal surface and must be excluded from the total surface area calculation.

Step-by-Step Process for Surface Area:

  1. List Visible Surfaces: Identify all the faces or curved surfaces that are exposed to the outside air.
  2. Calculate Exposed Areas: Use the relevant area formulas for each exposed surface.
    • Example: If a cube sits on top of a cuboid, the top face of the cuboid and the bottom face of the cube are hidden and must be ignored. You only calculate the curved surface area of a cylinder if the ends are covered by other shapes!
  3. Sum the Exposed Areas: Add only the areas of the external surfaces together.

Did you know?
If you attach a cylinder to a hemisphere, the joining area is a circle (\(A=\pi r^2\)). When calculating the surface area of the combined solid, you use the Curved Surface Area of the hemisphere (\(2\pi r^2\)) and the Curved Surface Area of the cylinder (\(2\pi rh\)), and then you add the bottom circular base of the cylinder (\(\pi r^2\)). The internal circle where they meet is ignored!


Part 4: Calculating Parts of Solids

This involves objects that are fractions of standard solids, like cutting a sphere in half or chopping off the top of a cone.

4.1 Hemispheres (Half a Sphere)

A hemisphere is half of a sphere. Radius \(r\).

  • Volume: Since it's half a sphere, we simply halve the sphere volume formula.
    \(V_{Hemisphere} = \frac{1}{2} \times V_{Sphere} = \frac{1}{2} \times (\frac{4}{3}\pi r^3) = \frac{2}{3}\pi r^3\)
  • Surface Area (SA): Be careful! A hemisphere has two surfaces:
    1. The Curved Surface Area (half the sphere's SA): \(\frac{1}{2} \times 4\pi r^2 = 2\pi r^2\).
    2. The Flat Circular Base: \(\pi r^2\).
    Total SA of Hemisphere = \(2\pi r^2 + \pi r^2 = 3\pi r^2\).

4.2 Frustum (Extended Content Hint)

A frustum is the part of a pyramid or cone that remains after the top part is cut off by a plane parallel to the base. If you see a bucket or a lampshade shape, you are dealing with a frustum of a cone.

Strategy for Frustum Volume and Surface Area:

  1. Volume: Calculate the Volume of the large (original) cone/pyramid. Calculate the Volume of the small (cut-off) cone/pyramid. Subtract the small volume from the large volume.
    \(V_{Frustum} = V_{Large} - V_{Small}\)
  2. Surface Area: This is more complicated, as you need three parts: Area of the bottom base + Area of the top base + Curved Surface Area of the frustum itself (which is the CSA of the large cone minus the CSA of the small cone).

Note: Frustum problems often rely on similar triangles to find the missing height or radius of the small cone/pyramid before calculations can begin.


Quick Review Box: Formulas Used in Mensuration (C6.4/E6.4)

The following formulas are usually given in Papers 1-4, but you must know how to apply them to composite shapes:

  • Area of a Circle: \(A = \pi r^2\)
  • Circumference of a Circle: \(C = 2\pi r\)
  • Volume of a Prism (or Cylinder): \(V = A l\) (Area of cross-section \(\times\) length)
  • Volume of a Pyramid (or Cone): \(V = \frac{1}{3} A h\) (Area of base \(\times\) height)
  • Curved Surface Area of a Cylinder: \(A = 2\pi r h\)
  • Curved Surface Area of a Cone: \(A = \pi r l\) (where \(l\) is the slant height)
  • Surface Area of a Sphere: \(A = 4\pi r^2\)
  • Volume of a Sphere: \(V = \frac{4}{3}\pi r^3\)

Summary: Your Compound Shapes Checklist

To succeed with compound shapes, always start by asking these three critical questions:

  1. Am I calculating Perimeter/Area (2D) or Surface Area/Volume (3D)? The rules are different.
  2. How can I decompose the shape? Split the complex shape into standard, measurable components.
  3. For Surface Area, did I exclude the internal faces? Remember the "Wrapping Paper" rule—only calculate surfaces that are exposed!

Practice makes perfect in Mensuration. Keep your formulas ready, work step-by-step, and you'll master these complex problems!