6.1 Geometric Properties of Three-Dimensional Shapes
Three-dimensional (3D) shapes possess length, width, and height/depth. They are categorized based on their faces, edges, and vertices:
- Prism: Has two congruent and parallel polygonal bases. Side faces are rectangles. Named after the shape of its base (e.g., triangular prism).
- Pyramid: Has one polygonal base and triangular side faces that meet at a single point called the apex.
- Cylinder: Has two congruent and parallel circular bases connected by a curved surface.
- Cone: Has one circular base, a slanted curved surface, and one apex.
- Sphere: A completely round 3D shape where every point on its surface is equidistant from its centre.
6.2 Nets of Three-Dimensional Shapes
A net is a 2D layout pattern formed by unfolding a 3D shape along its edges. When folded back, it reconstructs the 3D solid without overlapping.
6.3 Surface Area of Three-Dimensional Shapes
The total surface area of a 3D shape is the combined area of all its outer surfaces/faces.
Surface Area Formulae
- Prism:
$$\text{Surface Area} = 2 \times (\text{Base Area}) + \text{Sum of Areas of Rectangular Faces}$$
- Pyramid:
$$\text{Surface Area} = (\text{Base Area}) + \text{Sum of Areas of Triangular Faces}$$
- Cylinder:
$$\text{Surface Area} = 2\pi r^2 + 2\pi rh$$
- Cone: (where $s$ is the slant height, $s = \sqrt{r^2 + h^2}$)
$$\text{Surface Area} = \pi r^2 + \pi rs$$
- Sphere:
$$\text{Surface Area} = 4\pi r^2$$
6.4 Volume of Three-Dimensional Shapes
Volume is the amount of 3D space occupied by a shape.
Volume Formulae
- Prism / Cuboid / Cube:
$$\text{Volume} = \text{Base Area} \times \text{Height}$$
- Cylinder:
$$\text{Volume} = \pi r^2 h$$
- Pyramid:
$$\text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height}$$
- Cone:
$$\text{Volume} = \frac{1}{3} \pi r^2 h$$
- Sphere:
$$\text{Volume} = \frac{4}{3} \pi r^3$$
- Hemisphere:
$$\text{Volume} = \frac{2}{3} \pi r^3$$