Sectors, Surface Area & Volume

Measurement questions turn shapes into quantities: lengths, areas, surface areas, and volumes. The main challenge is deciding which part of the object is being measured. An arc is a length, a sector is an area, surface area covers the outside, and volume fills the inside.

CSEC Paper 02 often gives measurement questions with diagrams and real-world wording. Read the units carefully, identify whether the answer should be linear, square, or cubic units, and avoid rounding until the final line. Clear unit use is part of the explanation, not an extra decoration.

Arc Length¶

Arc length is a fraction of the full circumference. The central angle tells you what fraction of the circle you are using.

An arc is part of a circle's circumference.

$\text{Arc length} = \frac{\theta}{360°} \times 2\pi r$

Where $\theta$ = angle at center (in degrees).

Area of a Sector¶

A sector is a slice of a circle, so its area is the same fraction of the whole circle's area as its central angle is of 360°.

A sector is a "pizza slice" of a circle.

$\text{Sector area} = \frac{\theta}{360°} \times \pi r^2$

Area of a Segment¶

A segment is not the same as a sector. It is the curved cap left after removing the triangle formed by the two radii and the chord.

A segment is the area between a chord and the arc (NOT including the triangle).

$\text{Segment area} = \text{Sector area} - \text{Triangle area}$

---

Part 4: Surface Area of 3D Shapes¶

Surface area counts the outside faces only. Imagine unfolding the solid into a net and adding the areas of all exposed faces.

Surface area is the total area of ALL faces of a 3D shape.

Cube and Cuboid¶

Cube (all sides equal): $SA = 6s^2$

Cuboid (rectangular box): $SA = 2(lw + lh + wh)$

Prism¶

A prism has the same cross-section all the way through. Its surface area combines the two identical ends with the rectangular faces around the sides.

A prism has two identical parallel faces (bases) and rectangular sides.

$SA = 2 \times \text{(base area)} + \text{(perimeter of base)} \times \text{height}$

Cylinder¶

A cylinder's surface area comes from two circular ends plus one curved rectangle wrapped around the side.

$SA = 2\pi r^2 + 2\pi rh$

Or: $SA = 2\pi r(r + h)$

Where the first term is the two circular bases, the second is the curved surface.

Cone¶

A cone's surface area combines the circular base with the curved slant surface. Use slant height for surface area, not vertical height.

$SA = \pi r^2 + \pi rl$

Where $l$ = slant height (NOT the perpendicular height).

Sphere¶

A sphere has no flat faces, so its surface area uses a special formula based only on radius.

$SA = 4\pi r^2$

---

Part 5: Volume of 3D Shapes¶

Volume measures the space inside a solid, so answers use cubic units. Many formulas are built from the idea: area of base multiplied by height, with a fraction added for pointed solids.

Volume is the amount of space inside a 3D shape. It's measured in cubic units (cm³, m³, etc.).

Cube and Cuboid¶

For cuboids, volume counts layers of rectangular area stacked through the height. For cubes, all three dimensions are equal.

Cube: $V = s^3$

Cuboid: $V = l \times w \times h$

Prism¶

Any prism keeps the same cross-section throughout, so volume is cross-sectional area multiplied by length.

$V = \text{(base area)} \times \text{height}$

Cylinder¶

A cylinder is a circular prism. Its base area is $\pi r^2$, and the height tells how many circular layers are stacked.

$V = \pi r^2 h$

Pyramid¶

A pyramid has one third the volume of a prism with the same base area and height. This is why the formula includes $\frac{1}{3}$.

$V = \frac{1}{3} \times \text{(base area)} \times \text{height}$

Cone¶

A cone has one third the volume of a cylinder with the same base radius and height.

$V = \frac{1}{3} \pi r^2 h$

Sphere¶

Sphere volume depends on radius in three dimensions, so the radius is cubed. A small change in radius can make a large change in volume.

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