Grade 10 Area: Short Notes

10 ශ්‍රේණිය · 6 පරිච්ඡේදය · වර්ගඵලය

98%

of Grade 10 term tests include it (53 of 54)

596

marks in past papers (91 questions)

11 min

full lesson with quick checks

🎯 What you'll learn

Find the area of a sector using θ/360 × πr²
Find the area of a semicircle and a quadrant
Work backwards from a sector's area to find its radius or centre angle
Find the area of compound figures that contain sectors, by adding or subtracting parts

How much grass for the goat?

In a Kurunegala village, a goat is tied with a 14 m rope to the corner post where two fences meet at a right angle. It can only graze inside the fences. How many square metres of grass can it reach? The answer is a slice of a circle, and that slice is today's lesson.

Quick recap

Area is the amount of flat surface a shape covers, in cm² or m².• Rectangle: length × width• Triangle: ½ × base × height• Trapezium: ½ × (sum of parallel sides) × height• Circle: πr²In term tests take π = 22/7 unless told otherwise. Radii like 7, 14 and 21 make the working easy.

A sector is a slice of a circle

A sector is cut off by two radii and an arc. Its centre angle θ tells you what fraction of the whole circle it is: θ/360. Here θ = 60°, so the sector is 1/6 of the circle.

🧮 Area of a sector

Take the fraction θ/360 of the whole circle's area.Semicircle (180°): A = πr²/2Quadrant (90°): A = πr²/4

A = θ/360 × πr²

✏️ Back to the goat

The goat's rope is 14 m and the fences meet at 90°. Find the area of grass it can graze.

1The region is a quadrant: r = 14 m, θ = 90°, so it is ¼ of a circle.
2Whole circle = 22/7 × 14 × 14 = 616 m².
3Quadrant = ¼ × 616 = 154 m².

✅ 154 m²

⚠️ Arc formula or area formula?

Many candidates mix up 2πr (length round the edge) and πr² (surface inside). Ask yourself: is the answer in cm or cm²? Area needs r squared.Also: if the diameter is given, halve it first. With d = 14 cm, r = 7 cm and the semicircle's area is ½ × 22/7 × 7 × 7 = 77 cm², not 308 cm².

Working backwards

Sometimes the area is given and you must find r or θ.1. Write the formula: A = θ/360 × πr².2. Put in everything you know.3. Solve the equation.To find r you will reach r² = ..., so take the square root at the end. A radius is a length, so it is positive.

✏️ Find the centre angle

A slice of a round wattalappan tray is a sector of radius 14 cm with area 77 cm². Find its centre angle.

1Whole circle = 22/7 × 14 × 14 = 616 cm².
2θ/360 × 616 = 77, so θ/360 = 77/616 = 1/8.
3θ = 360 ÷ 8 = 45°.

✅ 45°

Compound figures: add or take away

Break the figure into shapes you know.• A sector attached to a rectangle: add the areas.• A sector cut out of a square: subtract it.First mark every length on the diagram. Equal radii, sides of a square and opposite sides of a rectangle often give you the length you need.

⚠️ Find the missing lengths first

Examiners report that many students start multiplying before they know the right lengths. In a compound figure the radius is often not written: it may equal a side of the square, or half the width. Mark r on the diagram, then calculate. And in the last part, use the area, not the perimeter, when a question says 'equal area'.

📌 Remember

• Sector area = θ/360 × πr² (take π = 22/7).• Semicircle πr²/2, quadrant πr²/4.• Area uses r² and units cm² or m²; arc length uses 2πr.• Diameter given? Halve it first.• Compound figure: mark all lengths, then add attached parts and subtract cut-out parts.

🧾 Quick revision sheet

Area of a circle	πr²
Area of a sector with centre angle θ	θ/360 × πr²
Area of a semicircle	πr²/2
Area of a quadrant	πr²/4
Area of a circle with r = 7 (π = 22/7)	154
Area of a circle with r = 14 (π = 22/7)	616
A 45° sector is what fraction of a circle?	1/8
Area of a ring between radii R and r	πR² − πr²