Arc lengths & sectors · Ages 14–15

Segments and composite figures

Segment = sector − triangle.

How to do it

Method

Segment = sector − triangle

90°, r = 6: 9π − 18 ≈ 10.26 cm²

Another way

Two quarters − the square

2 × 9π − 36 = 2 × (9π − 18)

In full

A segment is the part a chord cuts off: segment area = sector area − triangle area. Angle 90°, radius 6 cm: the sector is 14 × 36π = 9π, the triangle 6 × 6 ÷ 2 = 18, so the segment is 9π − 18 cm²; with π as 3.14, 28.26 − 18 = 10.26 cm². A composite figure: split it up or fill it in. The part of a square outside a quarter circle = square − quarter circle; the 'leaf' where two quarter circles overlap = 2 × segment, or think of it as two quarter circles − the square. A ring sector (a big sector less a small one) = θ360 × π(R² − r²). For a perimeter add every piece, each arc and each straight side, and leave out the lines inside.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
The window in the picture is a square of side 12 cm with a semicircle on top. Taking π as 3.14, what is the window's area, in cm²?
Show the answer and working
Answer: 200.52 cm²
Square 144; the semicircle's radius is 6: 12 × 3.14 × 6² = 56.52; 200.52 cm² in all.
Example 2
The sector in the picture has a radius of 16 cm and a right angle at the centre; the shaded part is a segment. The sector is π cm², the triangle cm², and the segment is the one less the other.
Fill in each blank
Show the answer and working
Answer: 64, 128
Sector 90360 × π × 16² = 64π; triangle 16 × 16 ÷ 2 = 128; segment = (64π − 128) cm².
Example 3
A semicircle has been cut out of the top side of the square in the picture, whose sides are 26 cm. Taking π as 3.14, what is the shaded part's perimeter, in cm?
Show the answer and working
Answer: 118.82 cm
Three straight sides 3 × 26 = 78, plus the semicircle's arc 3.14 × 13 = 40.82 (the side cut away is no longer on the edge): 118.82 cm.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.The square in the picture has sides of 3 cm; the quarter circle has its centre at one corner and the side as its radius. Taking π as 3.14, what is the shaded area, in cm²?
Answer:
2.The window in the picture is a square of side 4 cm with a semicircle on top. Taking π as 3.14, what is the window's area, in cm²?
Answer:
3.The sector in the picture has a radius of 20 cm and a right angle at the centre; the shaded part is a segment. The sector is π cm², the triangle cm², and the segment is the one less the other.
Fill in each blank
Answer:
4.A semicircle has been cut out of the top side of the square in the picture, whose sides are 20 cm. Taking π as 3.14, what is the shaded part's perimeter, in cm?
Answer:
5.The square in the picture has sides of 10 cm; the two quarter circles have their centres at opposite corners. Taking π as 3.14, what is the area of the shaded leaf, in cm²?
Answer:

All the answers

1. 1.935 cm²
2. 22.28 cm²
3. 100, 200
4. 91.4 cm
5. 57 cm²

Free worksheet: Segments and composite figuresWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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