Chords, arc length, and sectors
Chords, sectors, and segments divide circles into measurable pieces. Their formulas come from simple fractions of the whole circle and the Pythagorean theorem.
Chords and the centre
A perpendicular from the centre of a circle to a chord bisects the chord. Congruent chords are the same distance from the centre. A chord 16 cm long in a circle of radius 10 cm is the square root of (100 - 64) = 6 cm from the centre.
Finding a centre
The perpendicular bisector of any chord passes through the centre. Drawing two chords and their perpendicular bisectors locates the centre where the bisectors cross, a technique archaeologists use to reconstruct broken plates and wheels from fragments.
Intersecting chords
When two chords cross inside a circle, the products of their segments are equal: if one chord is cut into pieces 4 and 9 and the other into pieces 6 and x, then 6x = 36, so x = 6. Similar triangles formed by the chords prove this.
Area of a circle
A circle's area is \(A = \pi r^2\). A pizza of radius 15 cm has area \(225\pi \approx 707\) square centimetres. One way to see the formula: cut the circle into many thin slices and rearrange them into a near-rectangle about \(\pi r\) long and r high.
Sectors
A sector is a pizza-slice region bounded by two radii and an arc. Its area is the angle's fraction of the circle: \(A = \frac{\theta}{360}\pi r^2\). A 72-degree sector of a circle of radius 10 has area \(\frac{72}{360} \times 100\pi = 20\pi \approx 62.8\).
Segments
A segment of a circle is the region between a chord and its arc. Its area is the sector's area minus the triangle formed by the two radii and the chord. For a 90-degree sector of radius 6, that is 9 pi minus 18, about 10.3 square units.
Equation of a circle
On a coordinate plane, a circle with centre (h, k) and radius r has equation \((x - h)^2 + (y - k)^2 = r^2\), from the distance formula. Completing the square converts \(x^2 + y^2 - 6x + 4y - 12 = 0\) into \((x - 3)^2 + (y + 2)^2 = 25\): centre (3, -2), radius 5.
Worked example
A rotating lawn sprinkler sprays water 8 m and sweeps through 135 degrees. The watered area is 135/360 x 64 pi = 24 pi, about 75.4 square metres. The curved edge of the wet region is 135/360 x 16 pi = 6 pi, about 18.8 m long.
Key takeaways
A perpendicular from the centre bisects a chord, and congruent chords are equidistant from the centre; perpendicular bisectors of chords meet at the centre. Crossing chords have equal products of segments. Circle area is pi r squared. A sector's area is the angle's fraction of pi r squared, and a segment's area is a sector minus a triangle. A circle with centre (h, k) and radius r has equation (x - h) squared + (y - k) squared = r squared, found by completing the square.
Flashcard
From the centre, it bisects the chord
Check your understanding
A chord 16 cm long is in a circle of radius 10 cm. How far is it from the centre?