Question

Six circular biscuits of diameter 10 cm are arranged on a circular plate as shown below. What is the circumference of the plate in centimetres? 

Hint:

When dealing with problems involving circles packed together, connecting the centers of the circles is often the key. This usually reveals simple geometric shapes like equilateral triangles or regular hexagons, which simplifies the problem significantly.

Answer 1

Step 1: Understanding the Concept:
The problem asks for the circumference of a large circular plate that perfectly encloses six smaller, identical circular biscuits. To find the circumference of the plate, we first need to determine its radius.
Step 2: Key Formula or Approach:


The radius of each biscuit (\(r\)) is half of its diameter.
The centers of the six identical, touching circles form a regular hexagon.
The radius of the large plate (\(R\)) is the distance from the center of the arrangement to the outer edge of any biscuit.
The circumference of the plate is given by the formula \(C = 2 \pi R\).

Step 3: Detailed Explanation:


Calculate the biscuit radius: The diameter of a biscuit is 10 cm. \[ r_{\text{biscuit}} = \frac{\text{Diameter}}{2} = \frac{10 \text{ cm}}{2} = 5 \text{ cm} \]
Determine the geometry of the arrangement: The six biscuits are arranged symmetrically around a central point. The centers of these six biscuits form the vertices of a regular hexagon. The distance from the central point of the plate to the center of any biscuit is equal to the side length of this hexagon. Since the biscuits are touching, the side length of the hexagon is the sum of the radii of two adjacent biscuits, which is equal to the diameter of one biscuit. \[ \text{Distance from plate center to biscuit center} = \text{Diameter of biscuit} = 10 \text{ cm} \]
Calculate the radius of the large plate: The radius of the large plate (\(R\)) is the distance from its center to the outermost edge of one of the biscuits. This is the sum of the distance from the plate's center to a biscuit's center, and the radius of that biscuit. \[ R_{\text{plate}} = (\text{Distance from center to biscuit center}) + r_{\text{biscuit}} \] \[ R_{\text{plate}} = 10 \text{ cm} + 5 \text{ cm} = 15 \text{ cm} \]
Calculate the circumference of the plate: Now, use the formula for the circumference. \[ C = 2 \pi R_{\text{plate}} = 2 \pi (15) = 30\pi \text{ cm} \] Using the approximation \(\pi \approx 3.14159\): \[ C \approx 30 \times 3.14159 = 94.2477 \text{ cm} \]

Step 4: Final Answer:
The calculated circumference is approximately 94.25 cm, which falls within the accepted answer range of 92.9 to 94.6 cm.

Answer 2

Each biscuit has radius \( r = 5 \) cm. The six biscuits are arranged with their centres forming a ring around the centre of the plate, and each biscuit touches its two neighbours.

Consider the triangle formed by the plate's centre \( O \) and the centres of two neighbouring biscuits, \( C_1 \) and \( C_2 \). Since there are six evenly spaced biscuits, the angle \( \angle C_1OC_2 \) is \( \frac{360^{\circ}}{6} = 60^{\circ} \). Also, \( OC_1 = OC_2 \), since both biscuits sit the same distance from the centre, so this triangle is isosceles.

An isosceles triangle whose apex angle is \( 60^{\circ} \) must in fact be equilateral, since its other two angles are also equal and must sum with the apex to \( 180^{\circ} \), forcing all three angles to be \( 60^{\circ} \). So \( OC_1 = OC_2 = C_1C_2 \).

Since the two biscuits touch each other, \( C_1C_2 \) equals twice their radius, \( 2r = 10 \) cm, which means \( OC_1 = 10 \) cm as well.

The plate's radius is this distance plus one biscuit radius: \( R = 10 + 5 = 15 \) cm, so the circumference is

\[ C = 2\pi R = 2\pi(15) = 30\pi \approx 94.25 \text{ cm} \]

So the correct answer is 94.25 cm (approximately).