What angle does an arc of 10 cm subtend at the centre of a circle of radius 10 cm? Give the answer to the nearest 0.1°.

b) $\angle ADB$

The angle subtended by arc AB at the center is $\angle AOB = a$.

c) $\angle AOD$

d) $\angle ACD$

The angle subtended by arc AD at the center is $\angle AOD = 4a$.

e) $\angle BCD$

The arc BD has length $2a + 3a = 5a$.

f) $\angle BDC$

The arc BC has length $2a$.

g) $\angle CBD$

The arc CD has length $3a$.

h) $\angle CAD$

The arc CD has length $3a$.

Final Answers for :

a) $\boxed{\text{36}^\circ}$

b) $\boxed{\text{18}^\circ}$

c) $\boxed{\text{144}^\circ}$

d) $\boxed{\text{72}^\circ}$

e) $\boxed{\text{90}^\circ}$

f) $\boxed{\text{36}^\circ}$

g) $\boxed{\text{54}^\circ}$

h) $\boxed{\text{54}^\circ}$

:
Four pencils of diameter $7 \text{ mm}$ are held in a square formation by an elastic band.
The diameter of each pencil is $D = 7 \text{ mm}$. The radius is $r = D/2 = 3.5 \text{ mm}$.

The band forms a square around the four pencils. The length of each side of this square is equal to two diameters.
Length of one straight section $= 2D = 2 \times 7 \text{ mm} = 14 \text{ mm}$.
There are 4 such straight sections.
Total length of straight sections $= 4 \times 14 \text{ mm} = 56 \text{ mm}$.

At each corner, the band curves around a quarter of a pencil's circumference. There are 4 such quarter-circles.
The total length of the curved sections is equal to the circumference of one full pencil.
Circumference of one pencil $= \pi D = \pi \times 7 \text{ mm}$.

Total length = (Total length of straight sections) + (Total length of curved sections)

Using $\pi \approx 3.14159$:

If using $\pi \approx \frac{22}{7}$:

Final Answer for :
The length of the band is $\boxed{\text{78 mm}}$ (using $\pi = \frac{22}{7}$).

:
The elastic band is used to hold seven pencils as shown in Fig. 12.34. The diameter of each pencil is $7 \text{ mm}$.

The arrangement forms a hexagon. There are 6 straight sections.
Each straight section connects the centers of two adjacent pencils. The length of each straight section is equal to two radii, which is one diameter.
Length of one straight section $= D = 7 \text{ mm}$.
Total length of straight sections $= 6 \times 7 \text{ mm} = 42 \text{ mm}$.

At each of the 6 corners of the hexagon, the band curves around a pencil. The angle of each turn is $60^\circ$ (since it's a regular hexagon, the exterior angle is $360^\circ/6 = 60^\circ$).
So, each curved section is $\frac{60}{360} = \frac{1}{6}$ of the circumference of a pencil.
There are 6 such curved sections.
Total length of curved sections $= 6 \times \left(\frac{1}{6} \times \pi D\right) = \pi D = \pi \times 7 \text{ mm}$.

Total length = (Total length of straight sections) + (Total length of curved sections)

Using $\pi \approx 3.14159$:

If using $\pi \approx \frac{22}{7}$:

Final Answer for :
The length of the band is $\boxed{\text{64 mm}}$ (using $\pi = \frac{22}{7}$).

:
A piece of wire $22 \text{ cm}$ long is bent into an arc of a circle of radius $4 \text{ cm}$. What angle does the wire subtend at the centre of the circle?

Arc length $s = 22 \text{ cm}$.
Radius $r = 4 \text{ cm}$.

The formula for arc length is $s = r\theta$, where $\theta$ is the angle in radians.

To convert radians to degrees, multiply by $\frac{180^\circ}{\pi}$.

Using $\pi \approx \frac{22}{7}$:

Final Answer for :
The angle subtended at the center is $\boxed{\text{315}^\circ}$.

:
What angle does an arc of $10 \text{ cm}$ subtend at the centre of a circle of radius $10 \text{ cm}$? Give the answer to the nearest $0.1^\circ$.

Arc length $s = 10 \text{ cm}$.
Radius $r = 10 \text{ cm}$.

The formula for arc length is $s = r\theta$, where $\theta$ is the angle in radians.

To convert radians to degrees, multiply by $\frac{180^\circ}{\pi}$.

Using $\pi \approx 3.14159265$:

Rounding to the nearest $0.1^\circ$:

Final Answer for :
The angle subtended at the center is $\boxed{\text{57.3}^\circ}$.