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17.4 Semicircles and Sectors

GCSE Maths Foundation · Edexcel · Chapter 17: Perimeter, Area and Volume 2

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: area and perimeter of semicircles, quarter circles and sectors, and arc length. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Find the area and perimeter of a semicircle and a quarter circle
  • Write the angle of a sector as a fraction of a full turn (360°)
  • Work out the length of an arc and the area of a sector
  • Find the perimeter of a sector by adding the two radii to the arc

Key words

  • Semicircle: Half a circle, cut along a diameter.
  • Quarter circle: A quarter of a circle: a sector with a 90° angle.
  • Arc: Part of the circumference (a curved edge).
  • Sector: A “slice” of a circle between two radii and an arc.
  • Perimeter: The total distance around the outside: curved parts and straight parts.

Semicircles

A semicircle is half a circle, so its area is half of πr². Its perimeter is half the circumference plus the straight edge (the diameter).

Formulae: Area = ½ × π × r² Perimeter = ½ × π × d + d

Worked example: semicircle, diameter 10 cm

r = 5 cm. Area = ½ × π × 5² = 12.5π = 39.269… cm²
Curved edge = ½ × π × 10 = 15.707… cm. Perimeter = 15.707… + 10 = 25.707… cm

Answer: Area 39.3 cm², perimeter 25.7 cm

Quarter circles

A quarter circle is a quarter of the whole circle. Its perimeter has one curved edge and two straight edges, each a radius.

Worked example: quarter circle, radius 8 cm

Area = ¼ × π × 8² = 16π = 50.265… cm²
Arc = ¼ × 2 × π × 8 = 4π = 12.566… cm. Perimeter = 12.566… + 8 + 8 = 28.566… cm

Answer: Area 50.3 cm², perimeter 28.6 cm

Sectors as a fraction of a circle

A full turn is 360°. The angle at the centre tells you what fraction of the circle the sector is.

  • Fraction of circle; 180° 1/2; 90° 1/4; 120° 1/3; 60° 1/6; 45° 1/8; 270° 3/4

Formulae: Arc length = (θ ÷ 360) × 2 × π × r Sector area = (θ ÷ 360) × π × r² θ is the angle at the centre.

Worked example: arc length

A sector has radius 9 cm and angle 60°. Find the arc length.
60 ÷ 360 = 1/6. Arc = 1/6 × 2 × π × 9 = 3π = 9.424… cm

Answer: 9.42 cm (or exactly 3π cm)

Area and perimeter of a sector

Worked example: a sector with angle 135°

Fraction = 135 ÷ 360 = 0.375
Area = 0.375 × π × 10² = 37.5π = 117.809… cm²
Arc = 0.375 × 2 × π × 10 = 7.5π = 23.561… cm
Perimeter = arc + two radii = 23.561… + 10 + 10 = 43.561… cm

Answer: Area 117.8 cm², perimeter 43.6 cm

Worked example: a big sector (270°)

A sector has radius 4 cm and angle 270°. 270 ÷ 360 = 3/4.
Area = 3/4 × π × 4² = 12π = 37.699… cm²

Answer: 37.7 cm²

Common mistakes

  • Forgetting the straight edges. A semicircle's perimeter includes the diameter; a sector's includes two radii.
  • Using the diameter in πr² or in the arc formula with 2πr.
  • Dividing the angle by 180 instead of 360.
  • Mixing up arc length (a length, cm) with sector area (cm²).
  • Rounding the arc length before adding the radii.

Revision questions

Try these, then check your answers below each one.

Question 1. Work out the area of a semicircle with diameter 20 cm. Give your answer to 1 decimal place.

Answer: 157.1 cm²

Working: 1/2 × π × 10² = 50π = 157.079… = 157.1 cm².

Question 2. Work out the perimeter of a semicircle with diameter 20 cm. Give your answer to 1 decimal place.

Answer: 51.4 cm

Working: 1/2 × π × 20 + 20 = 31.415… + 20 = 51.4 cm.

Question 3. What fraction of a circle is a sector with angle 60°? (Choose from: 1/6, 1/4, 1/3, 1/5.)

Answer: 1/6

Working: 60 ÷ 360 = 1/6.

Related lessons

Continue in the full course

This lesson is section 17.4 (EGCMF1704) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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