Radius and diameter
A compass, a rope tied to a stake and the hand of a clock all measure a circle from the middle out, but a tape across a table, the size printed on a pizza box and the inside of a tin all measure it straight across. How do you get from one to the other, and what do they tell you about how circles fit in rows and boxes?
Where you are 15 in Circles
Why this exists
A square’s size is one number, the length of its side. A circle has no sides, but it too has one number that settles its size: how far its edge is from its center. Know that, and you know how wide the circle is, how far to open a compass to draw it, how long a rope must be to mark it out on the ground, and how big a box it needs.
The world hands you that size in two forms. A compass, a rope on a stake or a clock hand gives the distance from the middle out; a tape across a table or the inside of a box gives the distance all the way across. This lesson is about moving between the two, and about fitting circles into rows and boxes.
Definition
The words come from what a circle is. A radius is a segment from the center to a point on the circle, and its length is written r. A diameter is a chord through the center, and its length is written d. Each word names the segment and also its length, and the plural of radius is radii. A diameter is two radii laid end to end in one straight line, so
In everyday measurements, “across” means the diameter: a pizza 30 cm across, a cake tin 20 cm across. “From the center”, the opening of a compass and the length of a tether give the radius.
Intuition
Roll a coin between two rulers held parallel on a table. However you turn it, the rulers stay the same distance apart: a circle reaches exactly r beyond its center on each side, whichever way you look, so it is d wide in every direction. That is why a coin in a tray needs d of room each way, and a row of touching coins needs one diameter per coin.
Now push two coins together until they touch. The point where they meet sits on the straight line through their centers, so from one center to the other is the first coin’s radius and then the second’s. Two equal coins that touch have their centers one diameter apart.
Results
Result 1 (diameter and radius). For every circle, d = 2r and r = \dfrac{d}{2}.
Result 2 (the same width every way). A circle lies between two parallel lines that each touch it at one point, on opposite sides, and those lines are exactly d apart, whichever direction they run in. So a circle fits in a square of side d, touching all four sides, and in a box it needs d of room each way.
Result 3 (the compass). A compass opened to a gap of r draws a circle of radius r. To draw a circle d across, open it to \dfrac{d}{2}. A rope tied to a stake works the same way.
Result 4 (touching circles). When two circles with radii r_1 and r_2 touch, each outside the other, the point where they touch lies on the line through their centers, and the centers are r_1 + r_2 apart. Two equal circles that touch have their centers one diameter apart.
Why it is true
Result 1 is proved in what a circle is: the center cuts a diameter into two pieces, each a radius of length r, so d = r + r = 2r.
Result 2. Pick any direction and draw the diameter that points that way. At each of its ends, draw a line at right angles to it. The shortest way from a point to a line is straight across at right angles, so the nearest point of either line to the center is the end of the diameter, r away; every other point of the line is farther than r away and so outside the circle. Each line therefore touches the circle at one point, the whole circle lies between them, and they are d apart. A parallel line nearer the center crosses the diameter inside the circle and cuts through it; one farther out misses it. So two parallel lines that both just touch the circle are always d apart. Two directions at right angles give the four sides of a square of side d.
Result 3. A compass keeps one distance fixed. Its point stays on the center, and as the pencil swings round the gap never changes, so every mark is the same distance from the center. That is a circle, and the gap is its radius.
Result 4. Call the centers P and Q, and fold the page along the line PQ. A fold along a line through a circle’s center lands that circle on itself, because it keeps every distance from the center, so both circles land on themselves. If they touched at a point off the line, the fold would carry it to a second point where they meet, but circles that only touch meet at one point. So the touching point T is on the line, and since the circles are outside each other, it lies between the centers:
Worked examples
From one length to the other
(a) A coin has a diameter of 2.8 cm. Find its radius. (b) A round table has a radius of 0.85 m. Anwen wants a tablecloth that reaches at least from one edge of the table to the opposite edge. How wide must it be at least? (c) A pizza is 30\tfrac{1}{2} cm across. Find its radius.
Solution(a) The radius is half the diameter:
(b) From one edge to the opposite edge is the diameter, two radii:
The cloth must be at least 1.7 m wide.
(c) “Across” gives the diameter, so halve it. Half of 30 is 15 and half of \tfrac{1}{2} is \tfrac{1}{4}, so the whole number and the fraction can be halved separately:
Check. Doubling gives back the diameter: 2 \times 15\tfrac{1}{4} = 30 + \tfrac{1}{2}.
Marking out a garden bed
Eilidh marks out a round garden bed. She drives a stake S into the ground, ties a rope to it, and walks round with the rope pulled tight, scratching a line at the end of it. The rope is 1.9 m long from the stake to her hand. (a) She sets stones along the edge of the bed. How far is each stone from the stake? (b) Stones A and B are at opposite ends of a straight line through the stake. How far apart are they? (c) Next she wants a second round bed 3.4 m across. How long should the rope be?
Solution(a) The rope does what a compass does (Result 3): it keeps her hand 1.9 m from S the whole way round. So the edge is a circle with center S and radius 1.9 m, and every stone is 1.9 m from the stake.
(b) A and B are on the circle and the line between them passes through the center, so AB is a diameter:
(c) Now the diameter is given and the rope must be the radius, half of it:
A rope 3.4 m long would make a bed 6.8 m across, twice as wide as she wants, because the rope is a radius, not a diameter.
Coins in a tray
Cerys packs twelve identical coins into a tray in three rows of four. Each coin touches its neighbors, and the outside coins touch the sides of the tray. Inside, the tray is 8.8 cm long and 6.6 cm wide (Figure 1). Find the radius of each coin, and the distance between the centers of two neighboring coins.
Along the length, four coins sit side by side with no gaps, touching each other and both ends. By Result 2 each coin takes up one diameter of that length, so
The radius is half of that, r = 2.2 \div 2 = 1.1 cm. Two neighboring coins are equal circles that touch, so by Result 4 their centers are 1.1 + 1.1 = 2.2 cm apart, one diameter.
Check. Three coins fit across the width, and 3 \times 2.2 = 6.6 cm, the width of the tray.
Cutting discs from a strip
Olwen cuts round discs of radius 4.5 cm from a strip of card 9 cm wide and 40 cm long, side by side along the strip. (a) How many whole discs can she cut, and how long a piece of strip is left over? (b) What radius would let her cut exactly five discs from the same length of strip, with none left over?
Solution(a) Each disc has diameter d = 2 \times 4.5 = 9 cm. By Result 2 it needs 9 cm across the strip, which is exactly the width, and 9 cm along it. Four discs use 4 \times 9 = 36 cm of the length and five would need 45 cm, more than 40. So she cuts 4 discs, and 40 - 36 = 4 cm of strip is left over.
(b) Five discs side by side must use the whole 40 cm, so each diameter is
These smaller discs leave 9 - 8 = 1 cm of the strip’s width unused.
Drawing a row of circles
Faustino uses a compass to draw three circles with radii 1.5 cm, 2.5 cm and 3.5 cm. Their centers P, Q and R lie on one straight line, and each circle just touches the next (Figure 2). (a) How wide does he open the compass for each circle? (b) How long is the row, from the far left of the smallest circle to the far right of the largest? (c) How far is P from R?
(a) By Result 3 the gap between the compass point and the pencil is the radius, so he opens it to 1.5 cm, then 2.5 cm, then 3.5 cm.
(b) Along the line through the centers each circle covers one diameter (Result 2), and the circles meet with no gaps. So the row is the three diameters added:
(c) By Result 4 touching circles have their centers the two radii apart: PQ = 1.5 + 2.5 = 4 cm and QR = 2.5 + 3.5 = 6 cm. Q lies between the others on one line, so
Check. The row in (b) is PR plus the two outer radii sticking out beyond P and R: 10 + 1.5 + 3.5 = 15 cm.
Boundaries
“Across” must go through the center. A tape laid across a table that misses the center measures a chord, and every chord other than a diameter is shorter than d (what a circle is proves it). So the diameter of a round object is the widest reading across it.
Touching from outside. Result 4 is for circles that touch with each one outside the other. A small circle that touches a larger one from inside has its center r_2 - r_1 from the larger one’s center, not r_1 + r_2.
Straight rows only. The counts in the tray and strip examples are for circles in straight rows and columns. Rows shifted by half a circle, so that each circle sits in the dip between two below it, can sometimes fit more circles into the same box.
Width alone does not make a circle. Other curved shapes, built from arcs, are also the same width in every direction. Results 2 to 4 are about circles only.
Where this goes next
The radius and diameter are the two lengths every later circle fact is written in. Walking once round the circle gives the circumference, and the number that ties it to the diameter, \pi, is the subject of that lesson. The space inside is the area of a circle, written with the radius, and why that area formula works comes after it.
The lines in Result 2 that touch a circle at one point are tangents, studied in tangent properties. The compass is the main tool of compass and straightedge constructions, and on a coordinate grid the same fixed distance becomes the equation of a circle.
Exercises
From one length to the other
- A coin has a radius of 0.95 cm. Find its diameter.
- A round table is 1.35 m across through its center. Find its radius.
- A wheelbarrow wheel is 38 cm across. How high above flat ground is the center of the wheel?
- A clock face has a radius of 9\tfrac{3}{4} cm. Find its diameter.
- A circle’s diameter is 11 cm longer than its radius. Find the radius and the diameter.
- A circle’s radius and diameter add up to 45 cm. Find the radius and the diameter.
Compasses, ropes and hands
- Dov marks out a round garden bed with a rope 2.6 m long tied to a stake at the center. He pushes pegs into the ground along the edge. How far is each peg from the stake? Two of the pegs are at opposite ends of a straight line through the stake; how far apart are they?
- Faustino opens his compass to 6 cm, thinking that will draw a circle 6 cm across. How wide is the circle he actually draws, and how wide should he have opened the compass?
- Anwen measures from what she thinks is the center of a round table to three points on its edge and gets 70 cm, 70 cm and 64 cm. What can she conclude?
- On a round clock the minute hand reaches exactly from the center to the edge of the face and is 10.5 cm long. The hour hand is 7 cm long. Find the diameter of the clock face, and how far the tip of the hour hand is from the edge when it points at a number.
Rows and boxes
- Seven identical coins lie in a straight row, each touching the next, with their centers on one line. The row is 17.5 cm long from end to end. Find the radius of each coin.
- In the row of coins from Exercise 11, how far is the center of the first coin from the center of the last?
- A round pizza fits exactly in a square box, touching all four sides. The inside of the box is 36 cm wide. Find the radius of the pizza.
- A round cake has a radius of 13 cm. What is the smallest inside width of a square box it fits in?
- Round cookies 6 cm across are laid in a tray 50 cm long and 30 cm wide, in straight rows and columns like the coins in Example 3. How many cookies fit?
- Faustino draws a snowman as two circles, one on top of the other, with their centers on one upright line and touching each other. The radii are 2.5 cm and 4 cm. How tall is the snowman, and how far apart are the two centers?
- Two circles touch, each outside the other, and their centers are 9 cm apart. One has radius 3.5 cm. Find the radius and the diameter of the other.
Reasoning
- Explain why, to draw a circle of a given width with a compass, you open it to half that width.
- A round lid lies in a rectangular box and touches all four sides. Explain why the box must be a square, and how wide it is.
- A number n of equal circles of diameter d lie in a straight row, each touching the next, with their centers on one line. Explain why the row is nd long and why the first and last centers are (n - 1)d apart.
Answers
- 1.9 cm. d = 2r = 2 \times 0.95 = 1.9.
- 0.675 m. Across through the center is the diameter, and r = 1.35 \div 2 = 0.675.
- 19 cm. The ground is a straight line that touches the wheel at its lowest point, so by Result 2 the center is one radius above it: 38 \div 2 = 19.
- 19\tfrac{1}{2} cm. 2 \times 9 = 18 and 2 \times \tfrac{3}{4} = 1\tfrac{1}{2}, so d = 18 + 1\tfrac{1}{2} = 19\tfrac{1}{2}.
- r = 11 cm and d = 22 cm. The diameter is r + r, so it is longer than the radius by one radius: d - r = r = 11.
- r = 15 cm and d = 30 cm. r + d = r + 2r = 3r = 45, so r = 15.
- Each peg is 2.6 m from the stake, because the rope is a radius. The two opposite pegs are a diameter apart: 2 \times 2.6 = 5.2 m.
- The circle is 12 cm across, and he should have opened the compass to 3 cm. The opening is the radius, so d = 2 \times 6 = 12; for a circle 6 cm across he needs r = 6 \div 2 = 3.
- The point she chose is not the center, or the table top is not a circle, or a measurement is wrong. If the table were a circle and her point its center, all three distances would be radii and so all equal, and 64 \ne 70.
- The diameter is 21 cm, and the tip of the hour hand is 3.5 cm from the edge. The minute hand is a radius, so d = 2 \times 10.5 = 21; the hour hand falls short of the radius by 10.5 - 7 = 3.5.
- 1.25 cm. Each coin covers one diameter of the row, so 7d = 17.5 and d = 2.5; the radius is 2.5 \div 2 = 1.25.
- 15 cm. Neighboring centers are one diameter apart (Result 4), 2.5 cm, and there are six gaps between seven centers: 6 \times 2.5 = 15. Or take off the radius sticking out at each end: 17.5 - 1.25 - 1.25 = 15.
- 18 cm. The pizza touches two opposite sides of the box, and by Result 2 two parallel lines that both just touch a circle are exactly one diameter apart. So d = 36 and r = 36 \div 2 = 18.
- 26 cm. By Result 2 the cake needs one diameter of room each way, 2 \times 13 = 26, and a box 26 cm wide holds it touching all four sides.
- 40 cookies. Each cookie needs 6 cm each way. Along the length, 8 \times 6 = 48 \le 50 but 9 \times 6 = 54 > 50, so 8 fit in a row; across, 30 \div 6 = 5 rows fit exactly. 8 \times 5 = 40.
- The snowman is 13 cm tall and the centers are 6.5 cm apart. Up the line through the centers each circle covers one diameter: 2(2.5) + 2(4) = 5 + 8 = 13. By Result 4 the centers are the two radii apart: 2.5 + 4 = 6.5.
- Radius 5.5 cm, diameter 11 cm. By Result 4 the centers are r_1 + r_2 apart, so 3.5 + r_2 = 9, r_2 = 5.5 and d = 2 \times 5.5 = 11.
- A compass keeps its pencil a fixed distance from its point, so that gap is the radius of the circle it draws. The width of a circle is its diameter, twice the radius, so the radius, and the opening, is half the width.
- The lid touches both long sides, two parallel lines, so by Result 2 they are d apart: the box’s width is d. It touches both short sides too, so for the same reason its length is d. Length and width are equal, so the box is a square of side d, the lid’s diameter.
- Along the line through the centers each circle covers one diameter (Result 2), and touching circles leave no gap, so the row is n diameters, nd. Neighboring centers are one diameter apart (Result 4), and n centers have n - 1 gaps between them, so the end centers are (n - 1)d apart. This is also nd less the radius sticking out at each end: nd - \tfrac{d}{2} - \tfrac{d}{2} = (n - 1)d.