Perimeter is the total distance around a shape's edges, and area is the space it covers. For a rectangle, perimeter is 2(l + w) and area is l × w. For a circle, the edge is called the circumference: C = 2πr (or πd), and the area is A = πr². Composite figures are split into simple shapes, worked out separately, then added or subtracted.
Perimeter and area look easy, which is exactly why careless mistakes happen. The two most common ones: substituting the diameter into A = πr² when the question gave a diameter and the radius was never worked out, and forgetting that area is always in squared units while perimeter and circumference are in the base unit. On composite figures, students also forget to subtract a cut-out hole, or double-count a shared edge between two shapes.
The fix is not memorising more formulas. It is being disciplined about identifying every shape in the figure first, writing down radius or diameter clearly, and keeping units attached to every number as you work.
Rectangle
Perimeter = 2(l + w)
Area = l × w
Triangle
Perimeter = sum of 3 sides
Area = ½ × base × height
Circle
Circumference = 2πr = πd
Area = πr²
Composite figure
Split into simple shapes, then add or subtract
A circular garden has radius 7 m. Find its circumference and area, using π ≈ 22/7.
Solution
C = 2πr = 2 × 22/7 × 7 = 44 m.A = πr² = 22/7 × 7² = 22/7 × 49 = 154 m².Radius vs diameter: if the question gives you the diameter instead, halve it to get the radius before using A = πr². Circumference can use either form directly: C = πd or C = 2πr, but area always needs the radius.
A running track end is shaped like a rectangle 40 m by 20 m, with a semicircle of radius 10 m attached to one short side. Find the total area, taking π = 3.142.
Solution
40 × 20 = 800 m².πr², with r = 10: ½ × 3.142 × 10² = ½ × 3.142 × 100 = 157.1 m².800 + 157.1 = 957.1 m².A square of side 14 cm has a circle of radius 7 cm cut out of its centre. Find the area of the shaded region, using π ≈ 22/7.
Solution
14 × 14 = 196 cm².A = πr² = 22/7 × 7² = 154 cm².196 − 154 = 42 cm².The step students get wrong
On a shaded or "cut-out" region, students often add the two areas instead of subtracting, or subtract using the wrong shape's dimensions. Before calculating, write down in words what the shaded region actually is: usually "big shape minus small shape". Then compute each area separately and only subtract at the very last step, so an arithmetic slip in one part does not corrupt the other.
Unless a question tells you which value of π to take, use your calculator's π key or 3.142. Only use 22/7 when the question actually says to. On the Casio fx-97SG (both CW and X models), key the whole expression in one line, for example π × 7 x² for the area of a circle of radius 7, so the calculator rounds once at the end instead of at every step.
These formulas are the base layer for two related topics: once a figure becomes 3D, the same circle area formula reappears inside volume and surface area of solids (cones, cylinders and spheres all use πr² somewhere), and when a region is only part of a circle rather than the whole thing, the working extends into arc length and sector area and the area of a segment. All build directly on what is above.
Do I use radius or diameter for circle area?
Always radius for area: A = πr². If you are given the diameter, halve it first. Circumference can use either 2πr or πd directly, whichever matches the value you were given.
How do I know when to add areas and when to subtract?
Describe the shaded or required region in words first: if it is "the whole shape including a hole", add the parts; if it is "a shape with a piece removed", subtract. Writing this sentence before calculating prevents the most common composite-figure error.
What is the difference between perimeter and circumference?
They mean the same thing, the distance around the outside of a shape. "Circumference" is simply the specific term used for a circle's perimeter, and it uses 2πr or πd instead of adding straight sides.
What if the figure includes an arc instead of a full circle or semicircle?
A general arc needs the sector angle, not just the radius. See arc length and sector area for the fraction-of-360° method that extends this same circle formula.
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