Area and Perimeter of Composite Figures
Learn to find the area and perimeter of figures made of circles and other shapes
What are Composite Figures?
Composite figures are shapes made up of two or more simple shapes combined together. To find their perimeter or area, we need to:
- Identify the basic shapes that make up the figure
- Decide whether to add or subtract areas/lengths
- Calculate each part and combine them
For Perimeter
Identify all the outer edges - curved (arcs) and straight lines. Add them all up!
For Area
Either add areas (parts joined together) or subtract areas (parts cut out).
Key Strategies
Useful Combinations to Remember (for parts with the same radius)
2 semicircles
= 1 full circle
4 quarter circles
= 1 full circle
2 quarter circles
= 1 semicircle
2 quarter + 1 semicircle
= 1 full circle
Pro Tip: Always look for ways to combine parts of circles with the same radius into full circles or semicircles: this simplifies calculations! For perimeter, this joins the curved edges only. Still add every straight edge on the outside.
Worked Examples
Example 1: Quarter Circle in a Square
The figure shows a quarter circle in a square. Find (a) the perimeter and (b) the area of the shaded part.
A square of side 7 cm. A quarter circle of radius 7 cm is drawn from one corner of the square.
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Strategy:
The shaded part is the region outside the quarter circle but inside the square.
Solution:
Step 1: Find the length of the arc (quarter of circumference)
1/4 × 2 × 22/7 × 7
= 11 cm
Step 2: Find the perimeter of shaded part
11 + 7 + 7 (arc + two sides of square)
= 25 cm
Step 3: Find area of square
7 × 7
= 49 cm²
Step 4: Find area of quarter circle
1/4 × 22/7 × 7 × 7
= 38.5 cm²
Step 5: Find area of shaded part
49 − 38.5
= 10.5 cm²
The perimeter of the shaded part is 25 cm and the area is 10.5 cm².
Pro Tip: For perimeter, trace around the outside: you need the arc plus the two straight sides of the square. For area, subtract the quarter circle from the square.
Example 2: Two Overlapping Quarter Circles (Leaf Shape)
The figure is made up of two overlapping quarter circles. Find (a) the perimeter and (b) the area of the shaded part.
Two quarter circles of radius 7 cm overlap inside a square of side 7 cm. The leaf where they overlap is shaded.
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Strategy:
The perimeter is two equal quarter-circle arcs. For the area, the diagonal of the square cuts the leaf into two equal halves; each half is a quarter circle with a triangle taken away.
Solution:
Step 1: Find the length of one quarter-circle arc
1/4 × 2 × 22/7 × 7
= 11 cm
Step 2: Find the perimeter (2 arcs)
2 × 11
= 22 cm
Step 3: Find the area of one quarter circle
1/4 × 22/7 × 7 × 7
= 38.5 cm²
Step 4: Find the area of the triangle (half of the square)
1/2 × 7 × 7
= 24.5 cm²
Step 5: Find the area of half of the leaf (quarter circle − triangle)
38.5 − 24.5
= 14 cm²
Step 6: Find the area of the whole leaf (2 halves)
2 × 14
= 28 cm²
The perimeter is 22 cm and the area is 28 cm².
Pro Tip: Draw the diagonal of the square that joins the two tips of the leaf: it cuts the leaf into two equal halves, and each half is a quarter circle with a triangle taken away.
Example 3: Two Quarter Circles + Semicircle
The figure is made up of two quarter circles and a semicircle. Find the perimeter of the figure.
Two quarter circles stand on a semicircle. Each straight edge is 21 cm.
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Strategy:
The two quarter-circle arcs together are as long as one semicircle arc of the same radius. So the curved edges make one full circle.
Solution:
Step 1: Find the radius of the semicircle. Each straight edge is a radius of a quarter circle, and the two quarter circles meet in the middle, so the semicircle's diameter is 21 + 21 = 42 cm
42 ÷ 2
= 21 cm
Step 2: Recognise: 2 quarter circles + 1 semicircle of the same radius = 1 full circle
1/4 + 1/4 + 1/2
= 1 whole circle
Step 3: Find the circumference
2 × 22/7 × 21
= 132 cm
Step 4: Add the two straight edges
132 + 21 + 21
= 174 cm
The perimeter of the figure is 174 cm.
Pro Tip: Remember: 2 quarter circles + 1 semicircle of the same radius = 1 full circle. Look for these combinations to simplify your calculations!
Example 4: Two Semicircles + Quarter Circle
The figure is made up of 2 identical semicircles and a quarter circle. Find the area of the shaded part.
A quarter circle of radius 14 cm. A semicircle of diameter 14 cm is added on its top edge, and another is cut out of its left edge.
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Strategy:
One semicircle bulges OUT (adds area), one bulges IN (subtracts area). They cancel out, leaving just the quarter circle!
Solution:
Step 1: Find area of one semicircle (radius = 7 cm)
1/2 × 22/7 × 7 × 7
= 77 cm²
Step 2: Notice: +77 (bulging out) and −77 (bulging in) cancel out
77 − 77
= 0 cm²
Step 3: Shaded area = Quarter circle only (radius = 14 cm)
1/4 × 22/7 × 14 × 14
= 154 cm²
The area of the shaded part is 154 cm².
Pro Tip: When one shape adds area and another subtracts the same amount, they cancel out!
Example 5: Stadium Shape (Rectangle + Two Semicircles)
The figure is made up of a rectangle and two semicircles. Find the area of the figure.
A rectangle 100 m by 48 m with a semicircle on each of its two short sides.
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Strategy:
Area = Rectangle + 2 semicircles = Rectangle + 1 full circle
Solution:
Step 1: Find radius of semicircles
48 ÷ 2
= 24 m
Step 2: Find area of circle (2 semicircles = 1 circle)
3.14 × 24 × 24
= 1808.64 m²
Step 3: Find area of rectangle
100 × 48
= 4800 m²
Step 4: Find total area
4800 + 1808.64
= 6608.64 m²
The area of the figure is 6608.64 m².
Pro Tip: This stadium shape is used for running tracks! Break it into a rectangle plus two semicircles (which make one circle).
Example 6: Four-Point Star (Square minus 4 Quarter Circles)
The figure shows a square of side 8 cm. Four identical quarter circles are cut from its corners. Find the area of the shaded part.
One quarter circle is cut from each of the 4 corners of the square, leaving a 4-point star.
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Strategy:
Area = Square − 4 quarter circles = Square − 1 full circle
Solution:
Step 1: Find the radius of each quarter circle (half the side of the square)
8 ÷ 2
= 4 cm
Step 2: Find area of square
8 × 8
= 64 cm²
Step 3: Find area of 4 quarter circles (= 1 circle)
3.14 × 4 × 4
= 50.24 cm²
Step 4: Find area of shaded part
64 − 50.24
= 13.76 cm²
The area of the shaded part is 13.76 cm².
Pro Tip: Shortcut: 4 quarter circles = 1 full circle. Use this to calculate much faster!
Example 7: Two Semicircles (S-Shape)
The figure is made up of semicircles. Find the area and perimeter of the shaded part.
Two semicircles of diameter 14 cm side by side on one straight line, one above the line and one below it.
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Strategy:
Two semicircles = 1 full circle for area. Perimeter = 2 semicircle arcs + 2 diameters.
Solution:
Step 1: Find radius
14 ÷ 2
= 7 cm
Step 2: Find area (2 semicircles = 1 circle)
22/7 × 7 × 7
= 154 cm²
Step 3: Find arc length of each semicircle
1/2 × 2 × 22/7 × 7
= 22 cm
Step 4: Find perimeter (2 arcs + 2 diameters)
22 + 22 + 14 + 14
= 72 cm
The perimeter is 72 cm and the area is 154 cm².
Example 8: Two Circles and Two Lines
The figure shows 2 identical circles with two lines touching the circles. The radius of each circle is 10 cm. Find (a) the perimeter and (b) the area of the shaded parts.
Two circles of radius 10 cm touch each other. One straight line touches both circles at the top and one at the bottom. The shaded parts lie between the circles.
ShowHide solution
Strategy:
Together, the shaded parts are a square (between the two lines, from one centre to the other) minus two semicircles.
Solution:
Step 1: Find the length of each straight line (from above one centre to above the other)
10 + 10
= 20 cm
Step 2: Find the length of each curved edge (half of a circumference)
1/2 × 2 × 3.14 × 10
= 31.4 cm
Step 3: Find perimeter (2 semicircle arcs + 2 straight lines)
2 × 31.4 + 2 × 20
= 102.8 cm
Step 4: Find area of the square (each side is 20 cm: the two lines are one diameter apart)
20 × 20
= 400 cm²
Step 5: Find area of 2 semicircles (= 1 circle)
3.14 × 10 × 10
= 314 cm²
Step 6: Find shaded area
400 − 314
= 86 cm²
The perimeter of the shaded part is 102.8 cm and the area is 86 cm².
Key Takeaways
Break down composite figures into basic shapes you know how to calculate.
For perimeter: identify ALL outer edges (arcs + straight lines).
For area: decide if you need to ADD (shapes joined) or SUBTRACT (shapes cut out).
Look for simplifications (parts with the same radius): 2 semicircles = 1 circle, 4 quarters = 1 circle.
Always draw or visualise how parts can be rearranged to simplify the problem.