Circles: Practice

The two semicircular ends have the $20$ m width as their diameter, so each has radius $r = 10$ m. Two semicircles of equal radius make one full circle.

Rectangle area:

$A_{\text{rect}} = 50 \times 20 = 1000 \text{ m}^2$

The two ends together form a circle of radius $10$ m:

$A_{\text{circle}} = 3.14 \times 10^2 = 314 \text{ m}^2$

Add the pieces:

$A = 1000 + 314 = 1314 \text{ m}^2$

Start from $A = \pi r^2$ and solve for $r^2$ by dividing by $\pi$.

$r^2 = \frac{A}{\pi} = \frac{50.24}{3.14} = 16$

The radius is the number that squares to $16$:

$r = \sqrt{16} = 4 \text{ cm}$

The circle touches all four sides, so its diameter equals the side of the square, $10$ cm, giving radius $r = 5$ cm.

Square area:

$A_{\text{square}} = 10 \times 10 = 100 \text{ cm}^2$

Circle area:

$A_{\text{circle}} = 3.14 \times 5^2 = 78.5 \text{ cm}^2$

Subtract to find the leftover corners:

$100 - 78.5 = 21.5 \text{ cm}^2$

The flat bottom is the diameter, so the radius is half of it: $r = 3$ m. A semicircle is half a circle.

$A = \frac{1}{2}\pi r^2 = \frac{1}{2} \times 3.14 \times 3^2 = \frac{1}{2} \times 28.26 = 14.13 \text{ m}^2$

The outer circle (pond plus path) has radius $5 + 1 = 6$ m; the inner circle (pond) has radius $5$ m. The path is the area between them.

Outer area:

$A_{\text{outer}} = 3.14 \times 6^2 = 113.04 \text{ m}^2$

Inner area:

$A_{\text{inner}} = 3.14 \times 5^2 = 78.5 \text{ m}^2$

Subtract:

$113.04 - 78.5 = 34.54 \text{ m}^2$

Each full turn covers one circumference. With the diameter given, use $C = \pi d$.

$C = 3.14 \times 50 = 157 \text{ cm}$

Four turns cover four circumferences:

$4 \times 157 = 628 \text{ cm}$

The circle's circumference equals $44$ cm. Using $C = \pi d$, divide by $\pi$ to find the diameter.

$d = \frac{C}{\pi} = \frac{44}{3.14} \approx 14 \text{ cm}$

A quarter-circle is one fourth of a full circle, so take one fourth of $\pi r^2$.

The full circle area is $3.14 \times 8^2 = 3.14 \times 64 = 200.96 \text{ cm}^2$. One fourth of that is

$\frac{200.96}{4} = 50.24 \text{ cm}^2$

Use $A = \pi r^2 \approx 3 r^2$ and test the radius $4$ cm.

$A \approx 3 \times 4^2 = 3 \times 16 = 48 \text{ cm}^2$

A radius of $2$ cm gives about $12 \text{ cm}^2$ and a radius of $8$ cm gives about $192 \text{ cm}^2$, so $4$ cm is the match.

Replacing $r$ with $3r$ in the area formula gives

$A_{\text{new}} = \pi (3r)^2 = \pi \times 9 r^2 = 9 \pi r^2.$

The area is multiplied by $3^2 = 9$, because area depends on the radius squared.

Halve the diameter to get the radius: $r = 8$ in. Find the whole pizza's area, then divide by $8$.

$A = 3.14 \times 8^2 = 3.14 \times 64 = 200.96 \text{ in}^2$

One of eight equal slices is

$\frac{200.96}{8} = 25.12 \text{ in}^2$

Two semicircles of the same radius form one full circle of that radius, so just use $A = \pi r^2$ with $r = 7$ cm.

$A = 3.14 \times 7^2 = 3.14 \times 49 = 153.86 \text{ cm}^2$