Summary (Grade 10 NSC Matric Mathematics): Revision Notes

Summary

Understanding measurements in geometry

Area is the amount of space inside the boundary of a flat (2D) shape. It tells us how much surface a shape covers and is always measured in square units like cm², m², or mm².

Volume represents the amount of 3D space that an object occupies or the space inside a container. Volume is always measured in cubic units such as cm³, m³, or mm³.

Surface area is the total area of all the outer surfaces of a 3D object. Think of it as the amount of material needed to wrap the entire object.

Key geometric solids

Right prisms

A right prism is a 3D solid with two identical polygon bases connected by rectangular sides that are perpendicular (at right angles) to the base. The bases are parallel and the same shape and size.

Types of right prisms include:

• Triangular prism: has triangular bases
• Rectangular prism: has rectangular bases
• Cube: a special rectangular prism where all sides are equal length
• Cylinder: has circular bases (technically called a circular cylinder)

Pyramids and other solids

A pyramid is a 3D solid with a polygon base and triangular sides that meet at a single point called the apex. The sides are not perpendicular to the base.

A right pyramid has the line from the apex to the centre of the base perpendicular to the base.

A sphere is a perfectly round 3D shape where every point on the surface is the same distance from the centre.

A cone is similar to a pyramid but has a circular base instead of a polygon.

Area formulae for 2D shapes

These formulae help you calculate how much space flat shapes occupy:

• Square: $A = s^2$ (where $s$ = side length)
• Rectangle: $A = b \times h$ (where $b$ = base, $h$ = height)
• Triangle: $A = \frac{1}{2}b \times h$ (where $b$ = base, $h$ = perpendicular height)
• Parallelogram: $A = b \times h$ (where $b$ = base, $h$ = perpendicular height)
• Trapezium: $A = \frac{1}{2}(a + b) \times h$ (where $a$ and $b$ are parallel sides, $h$ = height)
• Circle: $A = \pi r^2$ (where $r$ = radius)

Volume formulae for 3D shapes

Prisms and cylinders

For prisms and cylinders, multiply the base area by the height:

• Rectangular prism: $V = l \times b \times h$
• Triangular prism: $V = \left(\frac{1}{2}b \times h\right) \times H$ (base area × height)
• Cube: $V = s^3$ (where $s$ = side length)
• Cylinder: $V = \pi r^2 \times h$

Pyramids, cones and spheres

These shapes have curved surfaces or come to a point:

• Square pyramid: $V = \frac{1}{3} \times b^2 \times H$
• Triangular pyramid: $V = \frac{1}{3} \times \frac{1}{2}bh \times H$
• Right cone: $V = \frac{1}{3} \times \pi r^2 \times H$
• Sphere: $V = \frac{4}{3}\pi r^3$

Surface area formulae

Surface area calculations require adding up all the exposed faces:

• Square pyramid: $SA = b(b + 2h_s)$ (where $h_s$ is slant height)
• Triangular pyramid: $SA = \frac{1}{2}b(h_0 + 3h_s)$
• Right cone: $SA = \pi r(r + h_s)$ (where $h_s$ is slant height)
• Sphere: $SA = 4\pi r^2$

Effects of scaling on measurements

When you multiply the dimensions of a shape by a scaling factor $k$:

• Area is multiplied by $k^2$
• Volume is multiplied by $k^3$
• Surface area is multiplied by $k^2$

This means if you double all dimensions ($k = 2$), the volume increases by $2^3 = 8$ times, while the surface area increases by $2^2 = 4$ times.

Worked examples

Let's calculate areas for these common shapes: