Areas Related to Circles

The One Thing

Areas of sectors are determined by the fraction of the full circle represented by the central angle. Areas of segments are obtained by combining sector, triangle, and whole-circle areas, with careful identification of whether the required region is minor or major.

Definitions and Results

• Circle: A plane figure whose every point is at the same distance from a fixed point called the centre.
• Radius: The line segment joining the centre of a circle to any point on its circumference.
• Diameter: A chord passing through the centre of a circle; its length is twice the radius.
• Chord: A line segment joining any two points on a circle.
• Arc: A part of the circumference of a circle.
• Central angle: The angle formed at the centre by two radii and subtending an arc.
• Sector: The region bounded by two radii and the arc between them.
• Minor sector: The smaller sector formed by two radii, usually having a central angle less than $180^{\circ }$.
• Major sector: The larger sector formed by two radii, usually having a central angle greater than $180^{\circ }$.
• Segment: The region bounded by a chord and the corresponding arc.
• Minor segment: The smaller region between a chord and its minor arc.
• Major segment: The larger region between a chord and its major arc.
• Circumference: The distance around a circle, calculated as

$$2\pi r.$$

• Area of a circle: The region enclosed by a circle, calculated as

$$\pi r^2.$$

• Area of a circle:

$$A=\pi r^2,$$ where $r$ is the radius.

• Circumference of a circle:

$$C=2\pi r.$$

• Length of an arc: For an arc subtending an angle $\theta$ at the centre,

$$\text{arc length}=\frac{\theta }{360^{\circ }}\times 2\pi r.$$

• Area of a sector: For a sector subtending an angle $\theta$ at the centre,

$$\text{area of sector}=\frac{\theta }{360^{\circ }}\times \pi r^2.$$ This requires $\theta$ to be measured in degrees and $r$ to be expressed in the relevant length unit.

• Alternative sector-area formula:

$$\text{area of sector}=\frac{1}{2}r\times \text{length of corresponding arc}.$$

• Area of a semicircle:

$$\frac{1}{2}\pi r^2.$$ Its curved boundary length is $$\pi r.$$

• Area of a quadrant: A quadrant is a sector of $90^{\circ }$, so

$$\text{area of quadrant}=\frac{1}{4}\pi r^2.$$

• Area of a minor segment:

$$\text{area of minor segment}=\text{area of corresponding sector}-\text{area of triangle formed by the two radii and the chord}.$$

• Area of a major segment:

$$\text{area of major segment}=\text{area of circle}-\text{area of minor segment}.$$

• Central angle of $180^{\circ }$: The sector is a semicircle, and the chord is a diameter.
• Triangle formed by two radii and a chord: This triangle is isosceles because both radii have equal length.
• Complementary segments: The sum of the areas of a minor segment and its corresponding major segment equals the area of the circle.
• Value of $\pi$: Use $\pi =\frac{22}{7}$ or $\pi =3.14$, depending on the instructions in the problem.
• Units: All measurements must use consistent units before a formula is applied.
• Composite regions: If a shaded figure consists of several parts, divide it into sectors, triangles, rectangles, or circles, then add or subtract their areas as appropriate.

Worked Methods

Method 1: Finding the area of a sector

• Identify the radius $r$ and central angle $\theta$.
• Confirm that the angle is measured in degrees.
• Calculate the fraction of the full circle represented by the sector:

$$\frac{\theta }{360^{\circ }}.$$

• Multiply this fraction by the area of the circle:

$$\text{area of sector}=\frac{\theta }{360^{\circ }}\times \pi r^2.$$

• Use the instructed value of $\pi$, either $\frac{22}{7}$ or $3.14$.

This method follows from the fact that a sector represents the same fraction of the circle’s area as its central angle represents of $360^{\circ }$.

Method 2: Finding the length of an arc

• Identify the radius $r$ and central angle $\theta$.
• Calculate the fraction of the circumference represented by the arc:

$$\frac{\theta }{360^{\circ }}.$$

• Multiply by the circumference $2\pi r$:

$$\text{arc length}=\frac{\theta }{360^{\circ }}\times 2\pi r.$$

Method 3: Finding the area of a sector using arc length

• Determine the radius $r$.
• Determine the length of the corresponding arc.
• Substitute into

$$\text{area of sector}=\frac{1}{2}r\times \text{arc length}.$$

• Check that the radius and arc length use consistent units.

Method 4: Finding the area of a minor segment

• Identify the minor sector formed by the two radii and the minor arc.
• Calculate the area of the sector:

$$\frac{\theta }{360^{\circ }}\times \pi r^2.$$

• Identify the triangle formed by the two radii and the chord.
• Calculate the area of this triangle.
• Subtract the triangle’s area from the sector’s area:

$$\text{area of minor segment}=\text{area of sector}-\text{area of triangle}.$$

The triangle is isosceles because its two sides are radii of the circle.

Method 5: Finding the area of a major segment

• Calculate the area of the whole circle:

$$\pi r^2.$$

• Calculate the area of the corresponding minor segment.
• Subtract the minor segment from the circle:

$$\text{area of major segment}=\text{area of circle}-\text{area of minor segment}.$$

This is generally more efficient than calculating the major sector and subtracting the corresponding triangle directly.

Method 6: Handling special sectors

• For a semicircle, use the central angle $180^{\circ }$:

$$\text{area}=\frac{1}{2}\pi r^2,\text{curved boundary length}=\pi r.$$

• For a quadrant, use the central angle $90^{\circ }$:

$$\text{area}=\frac{1}{4}\pi r^2.$$

• For a central angle of $180^{\circ }$, recognise that the chord is a diameter.

Method 7: Solving composite shaded regions

• Identify the required shaded region.
• Divide the figure into familiar shapes, such as sectors, triangles, rectangles, or circles.
• Calculate the area of each component.
• Add areas that form part of the required region.
• Subtract areas that are excluded.
• Check that all measurements use consistent units.

Where It Goes Wrong

• Using $\pi r^2$ for a sector without multiplying by $\frac{\theta }{360^{\circ }}$.
• Subtracting the triangle from the sector when the question asks for a major segment; the major segment is more efficiently found by subtracting the minor segment from the whole circle.
• Confusing a chord with an arc, or failing to identify that a chord passing through the centre is a diameter.
• Forgetting that a triangle formed by two radii and a chord is isosceles because the radii are equal.
• Applying formulas before converting measurements to consistent units.
• Selecting the wrong region in a composite shaded figure, rather than decomposing it into sectors, triangles, rectangles, or circles and adding or subtracting the relevant areas.

What Gets Asked

This material supports questions requiring students to:

• Define and distinguish circles, radii, diameters, chords, arcs, central angles, sectors, and segments.
• Calculate the circumference and area of a circle.
• Calculate the length of an arc for a given central angle.
• Calculate the area of a sector using

$$\frac{\theta }{360^{\circ }}\times \pi r^2$$ or $$\frac{1}{2}r\times \text{arc length}.$$

• Find the area of semicircles and quadrants.
• Find the area of a minor segment by subtracting a triangle from a sector.
• Find the area of a major segment by subtracting the minor segment from the whole circle.
• Recognise the special case of a $180^{\circ }$ sector as a semicircle with a diameter as its chord.
• Calculate shaded areas in composite figures by decomposing them into familiar shapes.
• Select and use $\pi =\frac{22}{7}$ or $\pi =3.14$ according to the instructions.
• Check the effect of consistent units and identify whether the required region is a minor or major sector or segment.