Shape and Space - Parts and Properties of Circles

If a circle has a radius of $7\text{ cm}$, what is its diameter?

$d = 2 \times r = 2 \times 7 = 14\text{ cm}$

Since the diameter is twice the length of the radius, we multiply the given radius ($7\text{ cm}$) by $2$.

Calculate the circumference of a circle with a diameter of $10\text{ cm}$ using $\pi = 3.14$.

$C = \pi \times d = 3.14 \times 10 = 31.4\text{ cm}$

Using the formula $C = \pi d$, we substitute $3.14$ for $\pi$ and $10$ for $d$ to get the perimeter of the circle.

A circular garden has a diameter of $14\text{ m}$. Find its radius and the distance around the garden (circumference) using $\pi = \frac{22}{7}$.

$r = \frac{14}{2} = 7\text{ m}$ $C = \pi \times d = \frac{22}{7} \times 14 = 22 \times 2 = 44\text{ m}$

First, we find the radius by dividing the diameter by $2$. Then, we calculate the circumference using the fractional value of $\pi$ to simplify the multiplication with the diameter.

Find the area of a circle with a radius of $3\text{ cm}$ (Take $\pi = 3.14$).

$A = \pi \times r^2 = 3.14 \times 3^2 = 3.14 \times 9 = 28.26\text{ cm}^2$

The area is calculated by squaring the radius ($3 \times 3 = 9$) and then multiplying the result by $\pi$ ($3.14$).

1. Find the radius: $r = \frac{d}{2} = \frac{20}{2} = 10\text{ cm}$.
2. Use the area formula: $A = \pi \times r^2$.
3. Substitute the values: $A = 3.14 \times (10)^2$.
4. $A = 3.14 \times 100 = 314\text{ cm}^2$.

To find the area, we must first convert the diameter into a radius. Since the radius is half the diameter, we use $10\text{ cm}$ in the area formula.

1. The distance traveled in one rotation is equal to the circumference ($C$).
2. Use the formula: $C = 2 \times \pi \times r$.
3. Substitute the values: $C = 2 \times \frac{22}{7} \times 35$.
4. Simplify: $C = 2 \times 22 \times 5$.
5. $C = 44 \times 5 = 220\text{ cm}$.

One full rotation of a wheel is equivalent to its circumference. We use the radius provided and the fraction value of $\pi$ to simplify the calculation.