Luas Lingkaran

$$\textbf{Area of a Circle}$$

The area of a circle is given by the following formula: $$A=\pi r^2$$ where $r$ is the radius of the circle.

What is a radius? It's the distance from the center of the circle to any point on the edge of the circle.

Here's more info on the radius of a circle.

$$\textbf{Solving for Area}$$

Sometimes you will be given a circle's radius and asked to find its area.

To solve this problem, we follow three steps.

Step One: Identify r, the Radius:

This one's easy! We are given in the question that the radius is $2 \text{ cm}.$

Step Two: Substitute r Into the Area of a Circle Equation :

$$\begin{align} A &= \pi r^2\\ &=\pi (2)^2\\ &=\pi (4)\\ &=4 \pi \end{align}$$

Step Three: Don't Forget the Units! :

Since we were given the radius in centimeters, the area will have the units of square centimeters, or $\text{cm}^2.$ The units for the area will always be the square of the units of the radius.

$$\textbf{Solving for Radius}$$

Sometimes, you will be given a circle's area and asked to determine its radius.

To solve this problem, we follow three similar steps as before:

Step One: Identify A, the Area:

We are given in the question that the area is $25\pi \text{ mm}^2.$

Step Two: Substitute A Into the Area of a Circle Equation :

$$\begin{align} A &= \pi r^2\\ 25\pi &=\pi r^2\\ 25\pi \cdot \frac{1}{\pi}&=\pi r^2 \cdot \frac{1}{\pi}\\ 25\cancel{\pi} \cdot \frac{1}{\cancel{\pi}}&=\cancel{\pi} r^2 \cdot \frac{1}{\cancel{\pi}}\\ 25 \cdot 1&= r^2 \cdot 1\\ 25&=r^2\\ \pm \sqrt{25}&=\sqrt{r^2}\\ \pm5&=r\\ r&=5, \text{ since the radius cannot be negative} \end{align}$$

Step Three: Don't Forget the Units! :

Since we were given the area in square millimeters, the radius will have the units of millimeters, or $\text{mm}.$

$$\textbf{Area of a Circle Picture Problems}$$

Sometimes you will be given a picture and asked to solve for a certain area. Here's an example where you are asked to find the area of the orange shaded region. (Note: Don't worry about including units.)

How do we find this area? First, we have to interpret the picture. Notice:

$$\text{area of orange region = area of big circle - area of small circle}$$

So, to find the area of the orange region ($A_O$) we first have to find the area of the small circle ($A_S$) and the area of the big circle ($A_B$).

Step One: Find the Area of the Small Circle:

Take a look at the small circle. The diagram shows that the diameter of this circle is $4.$ The radius is equal to half the diameter, so we can conclude that the radius of this circle is $4 \cdot \frac{1}{2}=2.$

Now we can substitute $r=2$ into our $A=\pi r^2$ equation in order to find the area of the small circle. $$\begin{align} A_S&=\pi r^2\\ &=\pi(2)^2\\ &=\pi (4)\\ &=4\pi \end{align}$$

So, $A_S=4\pi.$

Step Two: Find the Area of the Big Circle:

Next, we need to find the area of the big circle. From the diagram, we can see that the diameter of the big circle is equal to the length of the diameter of the small circle, which is $4,$ plus $12.$ Since the diameter of the big circle is $4+12=16,$ the radius is $16\cdot \frac{1}{2} = 8.$

Now we can substitute $r=8$ into our $A=\pi r^2$ equation in order to find the area of the big circle. $$\begin{align} A_B&=\pi r^2\\ &=\pi(8)^2\\ &=\pi (64)\\ &=64\pi \end{align}$$

So, $A_B=64\pi.$

Step Three: Solve for the Area of the Orange Region:

Remember: $$\text{area of orange region = area of big circle - area of small circle}$$ which we can rewrite as: $$A_O=A_B-A_S$$

Also remember: $$\begin{align} A_S&=4\pi\\ A_B&=64\pi \end{align}$$

Substituting $A_S=4\pi$ and $A_B=64\pi$ into our $A_O=A_B-A_S$ equation, we get: $$\begin{align} A_O&=A_B-A_S\\ &=64\pi - 4\pi\\ &=60\pi \end{align}$$

So, the area of the orange region, $A_O,$ is equal to $60\pi.$

$$\textbf{Area of a Circle}$$

The area of a circle is pi times the radius squared. $$A=\pi r^2$$

(Video) Area of a Circle

by mathematicsonline

Above is a video by mathematicsonline with a visual explanation of how to get this formula for the area of a circle.

$\large{\text{Summary}}$

The area of a circle is given by the formula $A=\pi r^2$

Where the area $A$ is given by the blue shaded region, $r$ is the radius of the circle, and $\pi\approx3.14$.

$$A_{\text{rectangle}} =\text{ base } \times \text{ height } = \pi r \times r$$

Here, the area of rectangle is exactly equal to the area of circle, so: $$A_{\text{circle}} = \pi r^2$$

(Video) Pi Day

by Expii

Ever wonder why the area of a circle is what it is? We can show you with some visually delicious math!

$\large{\text{Summary}}$

$\pi=\frac{\text{circumference}}{\text{diameter}}$

If you take a real pie you can see that the circumference divided by the diameter is pi. We also see that $\text{area} = \pi r^2$.

Take that pie and cut up the slices to form a rectangle. The width of the rectangle is $r$. The length of the side is half of the circumference. What is the area of the rectangle? It is $\text{length} \times \text{width}$ which is $\pi r^2$.