Circles: Circumference & Area
Calculate the circumference and area of circles using the formulae C = πd (or 2πr) and A = πr², and solve problems involving perimeters and areas of composite shapes that include circular parts
What a learner can do afterwards
- Calculate the circumference and area of a circle given its radius or diameter
- Find the perimeter or area of a composite shape made from rectangles and semicircles
- Explain the relationship between π, diameter, and circumference informally
1 · Read
You already know the parts of a circle. Now you can use two of them, the radius and the diameter, to measure a circle exactly. The distance around a circle is called its circumference. For any circle, the circumference is always about 3.14 times its diameter. That number is called pi, written as the symbol π. So the formula is C = πd, or if you know the radius instead, C = 2πr, since the diameter is twice the radius.
The space inside a circle is its area. The formula is A = πr², so you square the radius first, then multiply by π. Remember: area always uses the radius, never the diameter directly. If you are only given the diameter, halve it first.
A circular table has a radius of 3 cm. Its area is A = π × 3² = π × 9 ≈ 3.14 × 9 = 28.26 cm². Notice the radius gets squared before you multiply by π.
Composite shapes mix straight edges with circular ones. If you see a rectangle with a semicircle stuck on the end, work out the rectangle part and the semicircle part separately, then add them. A semicircle is exactly half of a full circle, so halve the circumference or area formula for that piece.
Circumference is π times the diameter (or 2πr), and area is π times the radius squared; break composite shapes into circles, semicircles, and straight parts, then add the pieces.
More to read
Ways to Find Perimeter
Perimeter is the distance all the way around the outside of a shape. Imagine building a fence around your yard. The length of that fence is the perimeter.
For most shapes, you just add up the lengths of all the sides. A rectangle with length 5 and width 3 has perimeter 5 + 3 + 5 + 3 = 16.
Circles are special because they have no straight sides. The distance around a circle has its own name: the circumference. To find it, use C = pi x d, where d is the diameter (the distance across the circle through its center). If you know the radius instead, use C = 2 x pi x r.
Pi (about 3.14) is the magic number that connects a circle's diameter to its circumference. The circumference is always a little more than 3 times the diameter, no matter how big or small the circle is.
For shapes made of parts, like a rectangle with a semicircle on top, add only the outside edges.
How to Find the Area of Irregular Shapes
Some shapes do not have their own formula, but do not worry. The trick is to break a tricky shape into smaller shapes you already know, like rectangles, triangles, and circles.
Here is the method. First, draw lines to split the big shape into simple parts. Second, find the area of each part using its formula: length x width for a rectangle, (base x height) / 2 for a triangle, and pi x r^2 for a circle. Third, add all the part areas together to get the total.
Sometimes it is easier to do the opposite: picture a big simple shape and subtract a piece that is missing. For example, a square with a circular hole has area = square area minus circle area.
This works for circular parts too. A running track shape might be a rectangle with a semicircle on each end. The two semicircles make one whole circle, so the total area is the rectangle area plus one full circle area. Split, solve, and add.
Ways to Find the Center of a Circle
The center of a circle is the most important point in it. Every point on the circle's edge is exactly one radius away from the center, and finding the center helps you measure the radius and diameter so you can calculate circumference and area.
One easy way is to draw crossed lines. Draw any chord (a line connecting two points on the circle). Find its midpoint and draw a line through it at a right angle. Do the same with a second chord. The two new lines cross at the center.
Another way is to fold. If you cut out a paper circle and fold it exactly in half, the fold line is a diameter. Fold it in half a different way, and the two fold lines cross at the center.
Once you know the center, measure from it to the edge. That is the radius, r. Double it to get the diameter. Then C = 2 x pi x r gives the distance around, and A = pi x r^2 gives the space inside.
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