KS3 · Maths
Perimeter and area of composite shapes (including circular parts)
A running track, an arched window, a pizza with a slice missing: none has its own formula, and you don't need one. Split it into pieces you know.
Maths · Sectors
Perimeter of a sector = arc + radius + radius. With radius 18 cm: at 40° the arc is 4π, so the perimeter is 4π + 36 (48.6 cm). At 80° the arc doubles to 8π, but the perimeter is only 8π + 36 (61.1 cm), which is less than double, because both radii stay 18 cm.
A sector with radius 18 cm. B slides round the circle, so the angle changes. Watch what the two radii do.
Maths · Composite area
Which move is the smart one?
Pick a shape, then pick the method that makes its area easiest to find.
Still to sort
Split and add (0)
The shape is made of separate parts side by side.
Where the line is: Nothing is cut out and nothing needs sliding about. You find each part's area and add them.
Take away (0)
A bigger shape has had pieces cut out of it.
Where the line is: If a piece has been cut out, you subtract its area. Adding the hole would be a slip.
Slide together (0)
The parts fit together into one simpler shape.
Where the line is: Use this when sliding the parts together gives a shape you can do in one go.
There are three ways to find the area of a composite shape. You are choosing, not calculating.
Composite perimeter
Problem
A closed shape has a straight bottom edge of 10 cm. At the right-hand end a quarter-circle arc of radius 3 cm curves up to the top edge. The top edge runs straight back to the left. At the left-hand end a semicircular arc of diameter 3 cm curves down to the bottom edge. The top edge is not labelled. Find the perimeter.
Maths · Algebra
Working backwards: from an area to a radius
Pick a shape and step through. Every line does the same thing to both sides.
Put the area into the area formula.
Step 1 of 3
Put the area into the area formula.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Split the shape into parts, find each one, and always go back to the full circle.
What you need to know
- A sector is the region between two radii and their connecting arc. An arc is part of a circle's circumference.
- Always split a composite shape into its component parts first, labelling any length two parts share. Then find each area or length and add, subtract or rearrange.
- A perimeter is the full distance round the outside: all the arcs plus all the straight segments.
- To find a radius from an area, rebuild the full circle first, then divide by π and take the square root.
The big picture
A composite shape is just its parts. Split it first, then find each area or length: add the parts, subtract the pieces that were cut out, or slide parts together into a simpler shape. The curved parts come from a full circle, so square the radius before taking the fraction. Perimeter is the whole distance round the outside, arcs plus straight edges, and a sector's two radii are part of it.
Key points
Worked example
Problem
A rectangle is 30 cm by 8 cm. A quarter circle of radius 8 cm has been cut out of one end. Find the area that is left, in terms of π and then to 1 decimal place.
⚠ Watch out
Forgetting to halve the diameter. If a question gives you the diameter, halve it to get the radius before you square anything. Squaring a diameter of 10 instead of a radius of 5 makes the area four times too big.
Memory hook
Full circle first, fraction second. Square the radius before you take the fraction, and turn a sector's area back into its full circle before you hunt for the radius.
Check yourself
A semicircle of diameter 20 cm sits on a rectangle 20 cm wide. What do you find first, and then what? (Radius 10 cm, square it, halve the circle, add the rectangle.)
Flashcards
(14)What is a sector?
What is an arc?
Radius or diameter: which is which?
A question gives you a diameter. What is your first move?
How do you find the area of part of a circle, such as a semicircle?
For a quarter circle, what do you divide by 4: the radius or the area?
Name the three ways to find a composite area.
What is the first step with any composite shape?
What is an arc length?
What is the perimeter of a sector?
What happens to a sector's arc and perimeter if you double the angle? If you double the radius?
How do you find the perimeter of a closed composite shape?
Do two shapes with the same area always have the same perimeter?
A semicircle's area is given. How do you find its radius?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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Keep me postedMore KS3 Maths topics
- Adding and subtracting positive and negative integers
- Algebraic notation and conventions
- Angle sum in a triangle
- Angles on parallel lines (corresponding, alternate, co-interior)
- Area of a circle
- Area of a trapezium
- Area of a triangle
- Area of composite rectilinear shapes
- Arithmetic sequences and finding the nth term
- Associative, commutative and distributive laws
- Bisecting an angle
- Calculating theoretical probabilities
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