Right Pyramids, Right Cones and Spheres (Grade 10 NSC Matric Mathematics): Revision Notes

Right Pyramids, Right Cones and Spheres

What are these 3D shapes?

Pyramids are three-dimensional shapes that have a polygon (flat shape) as their base and triangular faces that meet at a single point called the apex. The key feature is that the faces are not perpendicular to the base - they slope inward to meet at the top.

We name pyramids based on their base shape:

• Square pyramid: has a square base
• Triangular pyramid: has a triangular base

Right cones are similar to pyramids, but instead of a polygon base, they have a circular base. The apex sits directly above the centre of the circular base.

Spheres are perfectly round three-dimensional shapes that look the same from any direction. Every point on the surface is exactly the same distance from the centre.

Understanding key measurements

Before calculating surface areas and volumes, you need to identify these important measurements:

• Base area: The area of the bottom face
• Height (H): The perpendicular distance from apex to base
• Slant height: The diagonal distance from apex to the edge of the base
• Radius (r): Distance from centre to edge (for circles and spheres)

Surface area calculations

Surface area means the total area of all the faces or surfaces of a 3D shape. For pyramids and cones, this equals the base area plus the area of all the slanted sides.

Square pyramids

For a square pyramid with base side length $b$ and slant height $h_s$:

$\text{Surface Area} = b^2 + 4 \times \left(\frac{1}{2}bh_s\right) = b(b + 2h_s)$

The formula works because:

• $b^2$ is the area of the square base
• $4 \times \left(\frac{1}{2}bh_s\right)$ is the area of the four triangular faces

Triangular pyramids

For a triangular pyramid with base width $b$, base height $h_b$, and slant height $h_s$:

$\text{Surface Area} = \left(\frac{1}{2}bh_b\right) + 3 \times \left(\frac{1}{2}bh_s\right) = \frac{1}{2}b(h_b + 3h_s)$

This includes:

• $\left(\frac{1}{2}bh_b\right)$ for the triangular base area
• $3 \times \left(\frac{1}{2}bh_s\right)$ for the three triangular faces

Right cones

For a cone with radius $r$ and slant height $h$:

$\text{Surface Area} = \pi r^2 + \pi rh = \pi r(r + h)$

This combines:

• $\pi r^2$ for the circular base
• $\pi rh$ for the curved surface area

Spheres

For a sphere with radius $r$:

$\text{Surface Area} = 4\pi r^2$

This is a special formula that gives the total surface area of the curved surface.

Volume calculations

Volume measures how much space is inside a 3D shape. For pyramids and cones, the volume is always one-third of the base area multiplied by the height.

Key volume formulas

Square pyramid: $V = \frac{1}{3} \times b^2 \times H$

Triangular pyramid: $V = \frac{1}{3} \times \left(\frac{1}{2}bh_b\right) \times H$

Right cone: $V = \frac{1}{3} \times \pi r^2 \times H$

Sphere: $V = \frac{4}{3}\pi r^3$

Problem-solving strategy

Finding slant heights

Often you'll need to calculate the slant height using Pythagoras theorem:

For cones and pyramids: slant height² = height² + radius² (or half base width)

Exam tips and common mistakes