0580 · 5.3Circles, arcs and sectors

Circle measurements and formulas

0:00A curved garden border and the grass inside it require different measurements. The border needs a length, while the grass needs an area. If the garden is a slice of a circle, both measurements come from the same fraction of a whole turn. By the end, you will understand how that fraction connects circles, arcs and sectors. We need two starting ideas about measurement. Perimeter measures the distance around a shape, while area measures the space inside it.

0:24Lengths use units such as centimetres, and areas use square centimetres. We also need to remember that a complete turn contains three hundred and sixty degrees. A circle consists of points that are all the same distance from its centre. That distance is the radius. A diameter crosses the circle through its centre, so its length is twice the radius. The circumference is the distance all the way around the circle.

0:49Pi connects a circle’s circumference with its diameter. For every circle, circumference divided by diameter gives the same number, pi. Its decimal value begins three point one four, but that decimal does not end. This means circumference equals pi multiplied by diameter, whatever the circle’s size. Because the diameter is twice the radius, circumference also equals 2 × π multiplied by radius. Circle area equals pi multiplied by radius squared.

1:17These formulas measure different things, so their units must be different. Circumference is a length, while circle area is measured in square units. The area formula becomes more understandable if we imagine cutting a circle into many narrow sectors. Arrange these pieces alternately, pointing upwards and downwards. Their outline approaches a rectangle as the pieces become narrower. Its height approaches the radius, while its base approaches half the circumference. Half the circumference is pi multiplied by radius.

1:46Multiplying that base by the height gives pi multiplied by radius squared. This also explains why doubling the radius makes the area four times as large. Circumference only doubles, because its formula contains the radius rather than the radius squared. Our first exam style example gives a circle with diameter ten centimetres. We must calculate its circumference and area, giving exact answers in terms of pi. The diameter is already suitable for calculating circumference.

2:13For area, we first need the radius, so we divide ten by two to obtain five centimetres. Circumference equals pi multiplied by diameter. Substituting ten gives 10π centimetres. For the area, we substitute five into pi multiplied by radius squared. 5² = 25, so the area is 25π square centimetres. Both answers are exact because we have not replaced pi with a rounded decimal. A common mistake is putting the diameter into the area formula as though it were the radius.

2:44Here, that would produce 100π instead of 25π. The answer would be four times too large because the mistaken length is squared. Before substituting, identify whether the given measurement reaches across the whole circle or only from its centre.

Arc length and sector area

2:59An arc is part of a circle’s circumference. A sector is the region bounded by two radii and an arc. Its angle is measured at the centre of the circle. The key idea is that this angle tells us what fraction of the whole circle we have. A sector with an angle of ninety degrees occupies 90/360 of the circle. That simplifies to 1/4. Its arc takes this fraction of the circumference, and its area takes the same share of the circle’s area.

3:26The same fraction applies because both measurements belong to the same slice. For any sector, divide its central angle by three hundred and sixty to find the fraction of the circle. Multiply that fraction by the circumference to find arc length. For sector area, multiply it by the whole circle’s area instead. These are not unrelated formulas to memorise, but two applications of the same fraction. A larger central angle gives a longer arc and a larger sector area when the radius stays fixed.

3:53For example, doubling the angle doubles both measurements. Changing the radius has a different effect because area depends on radius squared. Always separate the effect of the angle from the effect of the circle’s size. Our second example has radius nine centimetres and central angle eighty degrees. We must calculate the arc length and sector area in terms of pi. First, the fraction is 80/360. Dividing the numerator and denominator by forty simplifies this to two ninths.

4:22The whole circumference is 2 × π × 9, giving 18π centimetres. Our arc occupies two ninths of that length. Dividing 18π by nine gives 2π. Multiplying that result by two gives an arc length of 4π centimetres. For the whole circle’s area, 9² gives eighty-one. Multiplying by pi gives 81π square centimetres. The sector occupies two ninths of this area. Dividing 81π by nine gives 9π, then multiplying by two gives 18π square centimetres.

Major sectors and reverse problems

4:57Two radii usually divide a circle into a smaller sector and a larger sector. The smaller one is the minor sector, with a central angle under one hundred and eighty degrees. The major sector is the larger region, and its reflex angle is greater than one hundred and eighty degrees. An angle of exactly one hundred and eighty degrees gives a semicircle. The two sector angles add to three hundred and sixty degrees.

5:18If the minor angle is one hundred and twenty degrees, the major angle is two hundred and forty degrees. Use the angle belonging to the region requested. A shaded major sector covers more than half the circle’s area, which provides a useful check. Our third example has radius six centimetres and a minor angle of one hundred and twenty degrees. We must calculate the major arc length, and the major sector’s area and perimeter, to three significant figures.

5:44Subtracting one hundred and twenty from three hundred and sixty gives a major angle of two hundred and forty degrees. Its fraction is 240/360, which simplifies to two thirds. The full circumference is 12π centimetres. Taking two thirds gives a major arc length of 8π centimetres. Using the calculator pi button gives twenty-five point one three two seven and further digits. Rounded to three significant figures, the arc length is twenty-five point one centimetres.

6:15The whole circle’s area is 36π square centimetres because 6² = 36. Two thirds of this gives 24π square centimetres. The calculator value begins seventy-five point three nine eight two. Our final sector area is seventy-five point four square centimetres to three significant figures. The sector perimeter includes its curved arc and both straight radii. Two radii of six centimetres contribute twelve centimetres altogether. Adding twelve to the unrounded arc length gives a value beginning thirty-seven point one three two seven.

6:47The perimeter is therefore thirty-seven point one centimetres to three significant figures. Some questions work backwards from a measurement to the angle. A sector has radius six centimetres and arc length 5π centimetres, and we must find its angle. The whole circumference is 12π centimetres, so the arc is 5π/12π of the circle. The pi cancels, leaving the fraction 5/12. Multiplying this fraction by three hundred and sixty degrees gives an angle of one hundred and fifty degrees.

7:17Arc length alone is not sector perimeter because it leaves out the straight boundaries. Another common error is using the minor angle for a major sector. Check the requested region before calculating its fraction. Also check your units, since an arc needs centimetres rather than square centimetres.

Exam methods and recap

7:34On Paper Two, these methods must work without a calculator. Simplify the angle fraction and carry out exact arithmetic, leaving pi when the question requests an exact answer. Paper Four allows a calculator, so use its pi button for decimal answers. Unless instructed otherwise, give answers that are not exact to three significant figures, without rounding intermediate values. The formulas are supplied in the formula list, but you still need to choose the correct measurements.

8:00Start by identifying the radius, then decide whether you need a whole circle or a fraction. For a sector, use its central angle over three hundred and sixty. Show that fraction in your working so your method is clear. Circumference measures the circular boundary, while area measures the space inside it. An arc and a sector take the same angle fraction of those whole circle measurements. Major sectors use the larger angle, and sector perimeter includes two radii.

8:26Understanding these connections lets you calculate accurately rather than guess which formula fits.