From the pages of your notebook to the grass on a soccer field, the area helps us measure the size of flat spaces in our everyday world.
In this guide, we’ll show you everything you need to know to understand and calculate area in math.
Read on for simple explanations, step-by-step examples, practice exercises, and answers to common questions students often ask!
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The area measures the amount of space a flat, or 2D, shape occupies. It helps us determine the amount of room inside a rectangle, square, or circle.
For example, imagine you’re drawing a big rectangle on a piece of paper and coloring it in. The area is the amount of space your crayon fills inside the rectangle’s edges.
In math, we use the symbol A to represent the area of a shape.
We measure the area using square units, like square inches (in2) or square centimeters (cm2), depending on our preferred units and metric system.
Why are the units squared?
It’s because area measures the amount of flat space a shape takes up and this space is two-dimensional.
A square unit has two dimensions—length and width—which is why we write the unit as 'in2,' with the exponent 2 showing that it is squared.
For example, in2 means an area is 1in long and 1in wide, while 2in2 means that an area is 2in long and 2in wide. In both cases, we can tell that we are dealing with squares as the length and width are equal in size.
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While there isn’t one universal rule for calculating the area of every 2D shape, we can start with something simple: finding the area visually.
Think of it like dividing a shape into square pieces, just like laying tiles on a floor. We count how many squares fit inside the shape to figure out its area.
Let’s look at a rectangle placed on a grid. The rectangle is made up of 12 grey squares, and each square is the same size as one grid square.
Since each grid square represents 1 square unit, the area of the rectangle is 12 square units.
Now, take a look at this triangle on the grid. It covers 6 full grey squares and 4 half-grey squares.
To find the area, we start with the 6 full squares, which count as 6 square units.
Next, we combine the 4 half squares—2 halves make 1 whole, so 4 half squares equal 2 full squares.
Adding these together, the triangle’s area is 6 + 2 = 8 square units.
Now, the visual approach—where we count squares inside a shape—works neatly for simple shapes like squares, rectangles, and even triangles.
But what about other shapes, like circles, pentagons, or weird blobs?
Counting squares for those might not be so easy!
Luckily, mathematicians came up with a smarter way: formulas.
Formulas are like shortcuts that help us find the area quickly, without having to divide every shape into tiny squares and count them one by one.
That’s exactly what we’ll explore in the next chapter!
Check out our video on how to find the area of basic 2D shapes!
A square is a shape where all four sides are the same length.
That makes things even easier for us.
To find the area of a square, we measure how much space it covers by multiplying the length of the side by itself.
So, the formula for the area of a square is:
A = s × s or simply A = s2
Here, s represents the length of one side of the square.
For example, if a square has a side length of 5 inches, its area is:
A = 5 × 5 = 25 in2
It’s that simple!
A rectangle is similar to a square, but instead of all four sides being the same, it has two longer sides (length) and two shorter sides (width). In other words, the two dimensions of this shape are length and width.
To find the area of a rectangle, we multiply the length by the width, or its one dimension by its other dimension.
The formula for the area of a rectangle is simple:
A = l × w
Here, l represents the length and w the width.
For example, if a rectangle is 8 inches long and 6 inches wide, its area is:
A = 8 × 6 = 48 in2
A triangle is different from a square or a rectangle because it has three sides and comes to a point.
Instead of having two equal dimensions, like a square, or a length and width, like a rectangle, a triangle has a base and a height.
The base (b) is any one of the triangle’s sides, and the height (h) is how tall the triangle is, measured straight up from the base to the top.
To find the area of a triangle, we use the formula:
A = \(\Large\frac{1}{2}\) × b × h
To find the area of a triangle, we multiply its two dimensions – the base by the height – and then divide by 2.
Why do we divide by 2?
Imagine you have a rectangle. To find its area, you multiply its length by its width.
Now, if you draw a line from one corner of the rectangle to the opposite corner, you’ll split the rectangle into two equal triangles like so:
The area of the rectangle is base × height.
Since the two triangles are equal and together make up the rectangle, each triangle must be half the area of the rectangle.
That’s why we use the formula:
A = \(\Large\frac{1}{2}\) × b × h
Now, let’s follow that up with an example.
Say a triangle has a base of 10 inches and a height of 6 inches, its area is:
A = \(\Large\frac{1}{2}\) × 10 × 6 = \(\Large\frac{1}{2}\) × 60 = 30 in3
Unlike squares, rectangles, or triangles, a circle is different because it is round and has no straight sides.
That means we can’t just multiply length and width to find its area. Instead, we use a special formula:
A = π × r2
Here:
r is the radius, which is the distance from the center of the circle to its edge.
π (pi) is a special number in math that never changes. It’s the number we get when we divide the circumference (the distance around a circle) by the diameter (the distance across a circle through the center). No matter how big or small a circle is, this ratio is always the same—about 3.14.
Why do we square the radius? – you might wonder.
Think of a circle as being made up of many tiny squares, all packed inside. The formula r2 tells us how many of those square units fit inside the circle
And why do we need in this formula?
Since a circle is round and not square, we use π to adjust the calculation and get the exact area. It helps us measure the space inside the circle correctly!
For example, if a circle has a radius of 4 inches, we calculate its area like this:
A = π × 42 = π × 16 ≈ 3.14 × 16 = 50.24 in2
So, the area of this circle is about 50.24 square inches!
We’ve explored how to find the area of squares, rectangles, triangles, and circles, but there are many more shapes out there!
While we can’t go over every single one, the principle remains the same—area measures how much space a shape covers, and we use a formula to calculate it.
Since each shape has its own formula, we’ve put together a table with the ones students usually encounter in their math class.
Now that we know the formulas for the area, let’s put them into action!
We’ll explore four examples—one for each shape we’ve learned about.
A basketball court is in the shape of a rectangle. If the court is 28 meters long and 15 meters wide, how much space does it cover?
We use the formula for the area of a rectangle:
A = l × w
Substituting with values:
A = 28 × 15 = 420 m2
So, the basketball court covers 420 square meters.
A new window is being installed in a classroom. If each side of the window is 4 feet long, what is its area?
Since a square has equal sides, we use the formula:
A = s2
Substituting the value:
A = 42 = 16 ft2
So, the area of the window is 16 square feet.
A playground has a triangular sandbox. The base of the triangle is 6 yards, and the height is 5 yards. What is its area?
We use the formula for the area of a triangle:
A = \(\Large\frac{1}{2}\) × b × h
Substituting the values:
A = \(\Large\frac{1}{2}\) × 6 × 5
A = \(\Large\frac{1}{2}\) × 30 = 15 yd2
So, the sandbox covers 15 square yards.
A round pizza has a radius of 7 inches. How much surface area does the pizza have?
We use the formula for the area of a circle:
A = π × r2
Substituting the values:
A = π × 72
A = π × 49 = 153.86 in2
So, the pizza has an area of about 153.86 square inches.
Now it’s your turn! Try these questions to test what you’ve learned about the area. Choose the correct answer for each one.
When you’re done, check your answers at the bottom of the guide.
Why do we measure area using square units?
A rectangular swimming pool is 10 meters long and 4 meters wide. What is its area?
Which formula do we use to find the area of a triangle?
A square has a side length of 6 inches. A rectangle has a length of 6 inches and a width of 5 inches. Which shape has the greater area?
A triangular garden has a base of 12 feet and a height of 5 feet. What is its area?
Here are some common questions students ask about the area, along with simple and clear answers!
Area measures how much space a shape covers, while perimeter measures the total distance around the shape.
For example, if you’re putting a fence around a garden, you need the perimeter. If you’re planting grass inside the garden, you need the area.
Area measures how much 2D a shape takes up, while volume measures how much space a 3D object fills.
Area is measured in square units (like cm2 or in2) because it has two dimensions (length and width).
Volume is measured in cubic units (like cm3 or in3) because it has three dimensions (length, width, and height).
For example, if you're painting a wall, you need to know the area. If you're filling a box with sand, you need to know the volume.
Not always! If you rearrange the shape without removing or adding space, the area stays the same. But if you stretch, shrink, or cut the shape, the area will change.
Yes! A rectangle and a triangle can have the same area, even if they look very different.
For example, a rectangle with a length of 8 cm and a width of 4 in has an area of:
A = 8 × 4 = 32 in2
A triangle with a base of 8 cm and a height of 8 cm also has an area of:
A = \(\Large\frac{1}{2}\) 8 × 8 = 32 in2
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If you’ve given our quick quiz a try, check your answers below.
Question 1: a. Because area measures space in two dimensions.
Question 2: c. 40 square meters
Question 3: b. A = \(\Large\frac{1}{2}\) b × h
Question 4: a. The square
Square A = 6 × 6 = 36 in2
Rectangle A = 6 × 5 = 30 in2
Question 5: b. 30 square feet
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