Perimeter is the total distance around the outside edge of a flat shape, and circumference is the perimeter of a circle. These measurements matter when you need to fence a yard, frame a picture, outline a track, or calculate the border of a design. In geometry, the key skill is identifying which edges are part of the outside boundary and adding only those lengths.
A clear diagram helps you separate the outer path from interior lines that should not be counted.
For polygons, perimeter is found by adding all exterior side lengths, while for circles the circumference depends on the radius or diameter. Composite figures combine straight segments with circular parts such as semicircles, quarter circles, and arcs. Arc length is a fraction of the full circumference, based on the central angle.
Careful labeling, consistent units, and checking whether a curved part uses radius or diameter will prevent most perimeter mistakes.
Key Facts
- Perimeter of a polygon: P = sum of all exterior side lengths.
- Circumference of a circle: C = 2πr.
- Circumference using diameter: C = πd.
- Diameter and radius relationship: d = 2r.
- Arc length with central angle θ in degrees: s = (θ/360)2πr.
- Semicircle arc length: s = πr, and quarter-circle arc length: s = (1/2)πr.
Vocabulary
- Perimeter
- Perimeter is the total distance around the outside boundary of a two-dimensional shape.
- Circumference
- Circumference is the distance around a circle.
- Radius
- Radius is the distance from the center of a circle to any point on the circle.
- Diameter
- Diameter is the distance across a circle through its center, equal to twice the radius.
- Arc Length
- Arc length is the distance along a curved part of a circle between two points.
Common Mistakes to Avoid
- Adding interior shared sides, which is wrong because perimeter includes only the outside boundary of the final shape.
- Using the diameter as the radius, which doubles the circumference or arc length and gives an answer that is too large.
- Counting a full circle when the diagram shows only a semicircle or quarter circle, which is wrong because only the visible curved boundary contributes to the perimeter.
- Mixing units such as centimeters and meters without converting first, which makes the sum meaningless because all lengths must use the same unit.
Practice Questions
- 1 A rectangle is 12 cm long and 5 cm wide. What is its perimeter?
- 2 A circle has radius 7 m. Find its circumference in terms of π, then approximate it using π = 3.14.
- 3 A composite figure is made by attaching a semicircle to one side of a rectangle. Explain why the shared side between the rectangle and semicircle is not included in the outside perimeter.