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How to find the area and the circumference of a circle

The two formulas: C = 2πr and A = πr²

For a circle of radius $r$ :

C = 2 π r (circumference - a length)

A = π r^{2} (area - a surface)

Telling them apart is easy if you look at the units. The circumference is measured in the same units as the radius (ft, m, cm), the area in square units (ft², m², cm²) - and that is exactly why the radius is squared in the second formula. If the task says to use $π = 3.14$ , use that value and do not round the result: the answer is expected to be exact for that $π$ .

Problem

Two worksheet tasks of the same type. Use $π = 3.14$ and do not round; include the units. Example 1. Find the circumference and the area of a circle with radius $2 ft$ . Example 2. Find the area and the circumference of a circle with radius $3 m$ .

The radius r runs from the centre to the edge. The circumference is the length of the boundary, the area is the surface inside it.

Answer

Radius

2 ft

C = 12.56 ft

A = 12.56 ft^{2}

. Radius

3 m

C = 18.84 m

A = 28.26 m^{2}

Checked by hand: both pairs recomputed with π = 3.14.

Step-by-step solution

Example 1: radius 2 ft

The radius is $r = 2 ft$ and we are told to take $π = 3.14$ . Circumference:

C = 2 π r = 2 \cdot 3.14 \cdot 2 = 12.56 ft

Area:

A = π r^{2} = 3.14 \cdot 2^{2} = 3.14 \cdot 4 = 12.56 ft^{2}

The two numbers came out equal - a coincidence that happens only at $r = 2$ , because there $2 r = r^{2}$ . The units still differ: feet against square feet, and mixing them up costs the mark.

Example 2: radius 3 m

A = π r^{2} = 3.14 \cdot 3^{2} = 3.14 \cdot 9 = 28.26 m^{2}

C = 2 π r = 2 \cdot 3.14 \cdot 3 = 6.28 \cdot 3 = 18.84 m

Here the coincidence is gone, and the area is the larger number - as it usually is once the radius passes $2$ .

How to check yourself

The area of a circle always sits between the inscribed square and the circumscribed one: between $2 r^{2}$ and $4 r^{2}$ . For $r = 3$ that is between $18$ and $36$ , and $28.26$ lands inside. For the circumference the quick check is $C \approx 6.28 r$ - a bit more than six radii.

Why the area formula works

Imagine cutting a circle into many narrow sectors and arranging them alternately, point up and point down. As the sectors get narrower, the shape approaches a rectangle. Its height approaches the radius $r$ ; its base approaches half the circumference, $π r$ . Therefore its area is

A = (π r) \cdot r = π r^{2} .

For a circle of radius $6 cm$ , the limiting rectangle has height $6 cm$ and base $6 π cm$ , giving area $36 π cm^{2}$ . Rearranging sectors does not change their total area.

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