Welcome to 3D Shapes and Volume!
Hi everyone! Get ready to explore the exciting world of three-dimensional shapes. Whether you are building a house, packing a suitcase, or drinking from a can, understanding 3D shapes and their volumes is essential in everyday life and, of course, for your IGCSE Maths exam!
This chapter is all about moving from flat (2D) shapes to solid (3D) objects. We will learn how to calculate the space *inside* these objects (Volume) and the total flat area *covering* the outside (Surface Area).
Section 1: The Fundamentals – Volume vs. Surface Area
What is Volume?
Volume is the measure of the space occupied by a 3D object. Think of it as how much stuff you can fit inside! Because we are multiplying three dimensions (length, width, height), volume is measured in cubic units (e.g., \(cm^3\), \(m^3\)).
- Analogy: If you are filling a swimming pool, you are measuring the volume of water it holds.
What is Surface Area (SA)?
Surface Area is the total area of all the faces (sides) covering the outside of the 3D shape. Since area is 2D, surface area is measured in square units (e.g., \(cm^2\), \(m^2\)).
- Analogy: If you are painting the outside of the swimming pool, you are calculating the surface area you need to cover.
Quick Tip: Always check the units required in the question! Volume is cubed (\(x^3\)) and Area is squared (\(x^2\)).
Section 2: Prisms and Cylinders (The "Stretching" Shapes)
A Prism is any 3D shape that has the exact same cross-section all the way along its length. If you slice it anywhere parallel to the base, the shape of the slice is identical.
General Formula for the Volume of Any Prism
This is the most important rule for this section! If you know the area of the front face (the cross-section), you just multiply it by how far back it stretches (the length or height).
Volume of Prism \(=\) Area of Cross-Section \(\times\) Length (or Height)
$$V = A_{cross} \times l$$
Cuboids and Rectangular Prisms
A cuboid is the simplest prism. Its cross-section is a rectangle.
- Volume (Cuboid): \(V = l \times w \times h\)
- Total Surface Area (Cuboid): Since there are three pairs of identical faces (front/back, top/bottom, left/right), you calculate the area of each unique face and double it.
$$SA = 2(lw + lh + wh)$$
Cylinders
A cylinder is a special type of prism where the cross-section is a circle.
1. Volume of a Cylinder
- The cross-section area (\(A_{cross}\)) is the area of the circle: \(\pi r^2\).
- We stretch this area by the height \(h\).
$$V = \pi r^2 h$$
2. Surface Area of a Cylinder
The total SA has three parts: two circular ends and the curved side.
- Area of the two ends: \(2 \times (\pi r^2)\)
- Area of the curved surface (Imagine peeling off the label! It's a rectangle with width \(h\) and length equal to the circumference of the circle, \(2\pi r\)): \(2\pi r h\)
$$SA_{Total} = 2 \pi r^2 + 2 \pi r h$$
The formula \(V = A_{cross} \times l\) is your best friend. Master finding the area of the cross-section (triangle, circle, trapezium, etc.) and the rest is just multiplication.
Section 3: Pyramids and Cones (The "Pointy" Shapes)
Pyramids and Cones are shapes that rise from a flat base to a single point (called the apex). They have a unique relationship with prisms and cylinders.
The Magic 1/3 Rule (Volume)
A pyramid (or cone) that fits exactly inside a prism (or cylinder) with the same base and height will have exactly one-third of the volume!
General Formula for Volume of Pyramid or Cone:
$$V = \frac{1}{3} \times \text{Area of Base} \times \text{Perpendicular Height}$$
$$V = \frac{1}{3} A_{base} h$$
Note: The height \(h\) used for volume must always be the perpendicular height (straight up from the center of the base to the apex).
Pyramids
If the base is a square or rectangle, \(A_{base} = l \times w\).
$$V = \frac{1}{3} lwh$$
Cones
The base is a circle, so \(A_{base} = \pi r^2\).
$$V = \frac{1}{3} \pi r^2 h$$
Surface Area of a Cone: Introducing Slant Height
To find the surface area of a cone, we need the slant height, usually labelled \(l\). This is the length measured along the slanted side from the base to the apex.
How to find \(l\)? If you look at the height (\(h\)), the radius (\(r\)), and the slant height (\(l\)), they form a right-angled triangle. We use Pythagoras' Theorem:
$$r^2 + h^2 = l^2$$
Formula for Total Surface Area of a Cone:
- Area of the Base (circle): \(\pi r^2\)
- Area of the Curved Surface: \(\pi r l\) (This is the area of the sector you get if you flatten the cone)
$$SA_{Total} = \pi r^2 + \pi r l$$
\(h\) (Perpendicular Height): Used for Volume and Pythagoras. (Straight up.)
\(l\) (Slant Height): Used for Curved Surface Area of cones/pyramids. (Diagonal side.)
Section 4: Spheres (The Round Shape)
A Sphere is a perfectly round 3D object, like a basketball or a globe. It is defined only by its radius (\(r\)).
Did you know? These two formulas are notoriously tricky to derive, so thankfully, you usually just need to know how to use them!
Volume of a Sphere
$$V = \frac{4}{3} \pi r^3$$
Memory Aid: Volume is cubed (\(r^3\)), and the number 4 is on top, 3 is on the bottom. V rhymes with four-thirds.
Surface Area of a Sphere
$$SA = 4 \pi r^2$$
Memory Aid: Area is squared (\(r^2\)), and the sphere covers the same area as 4 of its great circles.
Hemispheres (Half Spheres)
If you have a hemisphere, be careful!
- Volume (Hemisphere): Half the sphere volume. \(V = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3\)
- Total Surface Area (Hemisphere): This is the curved half (half of \(4\pi r^2\)) PLUS the flat circular base (\(\pi r^2\)).
$$SA_{Total} = 2 \pi r^2 + \pi r^2 = 3 \pi r^2$$
Section 5: Working with Formulas and Composite Solids
Solving for Missing Dimensions
Often, the question will give you the volume or surface area and ask you to find a missing side (like the radius or height). This is just algebra!
Example Step-by-Step: Finding the Radius
Problem: A cylinder has a volume of \(150\pi \ cm^3\) and a height of \(6\ cm\). Find the radius \(r\).
- Write down the formula: \(V = \pi r^2 h\)
- Substitute the known values: \(150\pi = \pi r^2 (6)\)
- Simplify the equation (Divide both sides by \(\pi\)): \(150 = 6 r^2\)
- Isolate \(r^2\) (Divide both sides by 6): \(r^2 = 25\)
- Find \(r\): \(r = \sqrt{25} = 5\ cm\)
Composite Solids (Shapes Stuck Together)
A composite solid is made up of two or more simple 3D shapes (e.g., a cone sitting on top of a cylinder, or a sphere with a hole drilled through it).
1. Calculating Volume of Composite Solids
This is straightforward. Calculate the volume of each component shape separately, and then add them together (or subtract if one shape is a void/hole).
2. Calculating Surface Area of Composite Solids (The Tricky Bit!)
When calculating the total surface area, you must only count the faces that are on the outside of the combined object.
- Common Mistake to Avoid: If a cone sits on a cylinder, you do not include the circular base of the cone or the circular top of the cylinder in your SA calculation, because they are touching and therefore inside the combined shape!
- The SA formula becomes: SA of Cone (Curved Part) + SA of Cylinder (Curved Part + Base).
V (Prism/Cylinder): \(A_{cross} \times l\)
V (Pyramid/Cone): \(\frac{1}{3} A_{base} h\)
V (Sphere): \(\frac{4}{3} \pi r^3\)
SA (Cylinder): \(2\pi rh + 2\pi r^2\)
SA (Cone, Curved): \(\pi r l\)
SA (Sphere): \(4 \pi r^2\)
Working with Exact Answers (\(\pi\) in the Answer)
In many exam questions, you will be asked to leave your answer "in terms of \(\pi\)." This means you treat \(\pi\) like an algebraic unknown and do not substitute the value 3.14... or use the \(\pi\) button on your calculator until the final step (if needed).
Example: Find the volume of a cylinder with \(r=3\) and \(h=5\), in terms of \(\pi\).
$$V = \pi r^2 h$$
$$V = \pi (3)^2 (5)$$
$$V = \pi (9) (5)$$
$$V = 45\pi$$
The final answer is \(45\pi\ cm^3\), not 141.37...
Keep practising these formulas, and remember that 3D problems often involve Pythagoras’ theorem to find those hidden heights and slant lengths! You've got this!