Mathematics (0580) | Surface area and volume

🚀 Ready to Conquer Surface Area and Volume (Mensuration)? 📐

Welcome to the chapter on Surface Area and Volume! This is one of the most practical sections in IGCSE Mathematics because it helps us measure the real world, from calculating how much paint we need for a room to figuring out how much water fits in a bottle.

Don't worry if three-dimensional shapes seem tricky at first. We will break down every shape into simple, manageable pieces. By the end of these notes, you'll be able to calculate the capacity and outer skin of cuboids, cylinders, cones, pyramids, and spheres! Let's get started!

1. The Difference: Volume vs. Surface Area

It is vital that you understand the difference between these two concepts. Think of a simple shoe box:

1.1 Volume (\(V\))

• What it is: The amount of three-dimensional space inside the solid. It measures capacity or how much the solid can hold.
• Analogy: How much water (or sand, or air) can fit inside the shoe box.
• Units: Always cubed (e.g., \(cm^3\), \(m^3\)).

🧠 Memory Aid: V-olume is V-ery much the inside.

1.2 Surface Area (\(A\))

• What it is: The total area of all the outer surfaces (faces) of the solid. It measures the "skin" of the object.
• Analogy: How much wrapping paper or paint you need to cover the outside of the shoe box.
• Units: Always squared (e.g., \(cm^2\), \(m^2\)).

🚨 Common Mistake to Avoid: Never confuse the units! Volume is \(^3\) and Area is \(^2\).

2. Prisms and Cylinders (The Uniform Shapes)

A Prism is any solid with a uniform cross-section. This means that if you slice it anywhere along its length, the cut shape remains the same. (Examples: a cuboid, a cylinder, a triangular chocolate bar box, a trapezoidal swimming pool.)

2.1 Volume of a Prism (The Golden Rule)

The formula for the volume of ANY prism is always the same:

$$V = A \times l$$

Where:

• \(A\) is the area of the cross-section (the uniform shape).
• \(l\) is the length or height of the prism.

Example: If the cross-section is a triangle (Area \(A = \frac{1}{2} \times base \times height\)), you multiply this area by the length of the prism to get the volume.

2.2 Cuboids (Rectangular Prisms)

A cuboid is the simplest prism.

• Volume: \(V = length \times width \times height\)
  (This is just \(V = A \times l\), where \(A\) is the base area \(l \times w\)).
• Surface Area: You must find the area of the 6 faces and add them up. Since opposite faces are identical:
  $$A = 2(lw) + 2(lh) + 2(wh)$$ (Twice the area of the top/bottom, plus twice the area of the front/back, plus twice the area of the two sides.)

2.3 Cylinders (Circular Prisms)

A cylinder is a prism where the cross-section is a circle (radius \(r\)).

Volume of a Cylinder (Formula Given)

Since the cross-section is a circle (\(A = \pi r^2\)) and the length is the height (\(h\)): $$V = \pi r^2 h$$

Surface Area of a Cylinder

The cylinder has three parts: the top circle, the bottom circle, and the curved side.

1. Area of the two ends (circles): \(2 \times (\pi r^2) = 2\pi r^2\)
2. Curved Surface Area (CSA): Imagine peeling the label off a can—it unrolls into a rectangle. The width of the rectangle is the height (\(h\)), and the length is the circumference of the circle (\(2\pi r\)).

   Formula (Given): $$A_{curved} = 2\pi r h$$

3. Total Surface Area: $$A_{total} = 2\pi r h + 2\pi r^2$$

Key Takeaway for Prisms: If you can find the area of the cross-section, you can find the volume! For SA, count and calculate the area of all the visible surfaces.

3. Pyramids and Cones (The Pointed Shapes)

Pyramids and cones come to a point (apex). They share a special relationship with their corresponding prisms and cylinders.

3.1 Volume of Pyramids and Cones

Any pointed solid that fits perfectly inside a prism (with the same base and height) has exactly one-third the volume of that prism.

Volume of a Pyramid (Formula Given)

$$V = \frac{1}{3} A h$$ Where \(A\) is the area of the base, and \(h\) is the perpendicular height (not the slant height!).

Volume of a Cone (Formula Given)

A cone is a circular pyramid. The base area \(A = \pi r^2\). $$V = \frac{1}{3} \pi r^2 h$$

Did you know? This 1/3 rule was proven mathematically by ancient Greek scholars! It’s a great example of the connection between maths and the physical world.

3.2 Surface Area of Pyramids and Cones

To find the total surface area of these shapes, you must often use Pythagoras' Theorem (\(a^2 + b^2 = c^2\)) to find the slant height (\(l\)).

Cones

If the cone has radius \(r\) and perpendicular height \(h\), the slant height \(l\) is the hypotenuse of the right-angled triangle formed inside:

$$l^2 = r^2 + h^2$$

The total SA of a cone includes the base (circle) and the curved surface.

1. Base Area: \(\pi r^2\)
2. Curved Surface Area (CSA) (Formula Given): $$A_{curved} = \pi r l$$
3. Total Surface Area: $$A_{total} = \pi r l + \pi r^2$$

Pyramids

The SA of a pyramid is NOT given as a single formula, so you must calculate the areas of the individual faces:

$$A_{total} = Area \ of \ Base + Area \ of \ all \ triangular \ faces$$ (Remember that the faces are triangles, area \( = \frac{1}{2} \times base \times slant \ height\). You may need to use Pythagoras' theorem to find the slant height for the triangular faces!)

Key Takeaway for Pointed Solids: Volume is always one-third of the related prism volume. Surface Area calculations often require finding the slant height first.

4. Spheres

A sphere is a perfectly round 3D object, like a basketball. It is defined only by its radius (\(r\)).

4.1 Volume of a Sphere (Formula Given)

$$V = \frac{4}{3} \pi r^3$$ (Notice the power of 3, matching the cubic units for volume!)

4.2 Surface Area of a Sphere (Formula Given)

$$A = 4 \pi r^2$$ (Notice the power of 2, matching the square units for area!)

5. Compound and Parts of Solids (Extended/Advanced Skills - C6.5 / E6.5)

This is where we combine the shapes we learned above, or deal with only a fraction of a solid.

5.1 Compound Solids

A compound solid is two or more simple solids joined together (e.g., a hemisphere on top of a cylinder, or a cone attached to a cuboid).

Calculating Volume:
This is the easiest part! You simply calculate the volume of each component shape separately and add them together.

Calculating Surface Area:
This is trickier because you only calculate the area of the surfaces that are exposed (the ones you could touch).

Step-by-Step SA for a Compound Solid:

1. Identify which surfaces are touching and therefore not exposed. (Example: If a hemisphere sits exactly on top of a cylinder, the circular base of the hemisphere and the circular top of the cylinder are hidden—you must subtract their area from the calculation.)
2. Calculate the exposed area of the first shape.
3. Calculate the exposed area of the second shape.
4. Add the exposed areas together.

5.2 Parts of Solids

Hemispheres (Half a Sphere)

A hemisphere is half a sphere.

1. Volume: Half the volume of a full sphere: $$V_{hemi} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3$$
2. Surface Area: This requires two parts:
   • Half the surface area of the sphere (the dome): \(\frac{1}{2} \times 4\pi r^2 = 2\pi r^2\)
   • The circular base (the flat part): \(\pi r^2\)
   $$A_{total \ hemi} = 2\pi r^2 + \pi r^2 = 3\pi r^2$$

🚨 Common Mistake: Students often forget to include the flat circular base when calculating the total surface area of a solid hemisphere!

Frustums (Extended Content E6.5)

A frustum is the part of a cone or pyramid that remains when the top section (a smaller cone or pyramid) is sliced off by a plane parallel to the base.

Step-by-Step Method (Involving Similarity):

1. Use the dimensions given and the concept of similar shapes to find the height and slant height of the small, cut-off cone/pyramid.
2. Volume of Frustum: Calculate the Volume of the Large Solid minus the Volume of the Small Solid. $$V_{frustum} = V_{large} - V_{small}$$
3. Surface Area of Frustum: Calculate the Area of the Large Solid's curved surface/faces minus the Area of the Small Solid's curved surface/faces, and then ADD the areas of the two bases (the large base and the new, smaller top face).

Key Takeaway for Compound Solids: Be careful with Surface Area—identify the hidden surfaces and only calculate what is exposed!

✅ Final Checklist and Formulas

Make sure you know how to use these formulas (which are provided in your examination, but speed is key!):

Formulas Provided in Exam (C6.4 / E6.4)

• Volume of a Prism: \(V = A l\) (where \(A\) is cross-sectional area)
• Volume of a Pyramid: \(V = \frac{1}{3} A h\) (where \(A\) is base area)
• Cylinder Volume: \(V = \pi r^2 h\)
• Cylinder Curved SA: \(A = 2\pi r h\)
• Cone Volume: \(V = \frac{1}{3} \pi r^2 h\)
• Cone Curved SA: \(A = \pi r l\)
• Sphere Volume: \(V = \frac{4}{3} \pi r^3\)
• Sphere Surface Area: \(A = 4\pi r^2\)

Formulas You Need to Construct/Know

• Cuboid Volume: \(V = l w h\)
• Cuboid Surface Area: Sum of 6 rectangular faces (e.g., \(2(lw) + 2(lh) + 2(wh)\)).
• Total Cylinder SA: \(A = 2\pi r h + 2\pi r^2\)
• Right-angled Connection (for Slant Height): \(l^2 = r^2 + h^2\) (Pythagoras)

Keep practising these calculations and always write down the formula before substituting your values. You got this!