Area
Understand the concept of area as a 1x1 grid measurement, learn the formulas, and practice calculating shapes.
Area Study Guide
Understand the concept of area as a 1x1 grid measurement, learn the formulas, and practice calculating shapes.
Area Practice Topics
1. Square & Rectangle
Learn and practice calculating the area of squares, rectangles, composite shapes, and shapes with holes.
Area Study Guide
The area of a shape is the amount of space inside its boundary. It is measured in square units, representing how many 1 by 1 unit squares can fit inside the shape.
1. What Are a Rectangle and a Square?
A rectangle is a flat (2D) shape with 4 sides and 4 right angles (90°). Its opposite sides are equal and parallel — one pair of sides is called the width (w) and the other is called the height (h). When all four sides are different, it is simply called a rectangle.
A square is a special rectangle where all 4 sides are equal in length. Because it is a rectangle, it also has 4 right angles. We call its side length s. You can think of a square as a rectangle where w = h = s.
2. Area as a 1 × 1 Grid
Imagine drawing a grid of 1 by 1 squares over a shape. Counting the total number of grid squares gives the area. For a rectangle with a width of 5 units and a height of 6 units, you can draw a grid of 5 columns and 6 rows. Counting them yields exactly 30 squares, so the area is 30 square units.
3. Rational and Irrational Area
The side lengths of a shape can be any real number: either rational (like whole numbers, decimals, or fractions) or irrational (such as square roots like ). If side lengths are irrational, the area itself might be irrational or rational. To read more about classifications like integers, rational, real, or complex numbers, refer to the Real & Complex Number Sets topic.
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Instead of counting grid squares, we use mathematical formulas:
• Rectangle: The area is the product of its width and height: A = w × h.
• Square: Since a square is a rectangle with w = h = s, we substitute into the rectangle formula and get: A = s × s = s².
Mastering SealMath: Entering the Square Root
To enter a square root (like ) in the math input, you have two options:
• Keyboard shortcut: Type \sqrt in the input box (backslash + the word sqrt). MathLive will instantly create the root symbol with the cursor inside — then just type your number.
• Virtual keyboard: Click the ⌨️ keyboard icon inside the input box to open the on-screen keyboard, then press the √□ button found on the 123 tab (second row, far right).
Frequently Asked Questions
How is the area of a shape defined?
The area of a shape is defined by how many unit squares of 1 by 1 fit inside it. For example, if a rectangle can be divided exactly into 30 squares of 1 by 1, its area is 30.
How do you calculate the area of a rectangle and a square?
The area of a rectangle is calculated as width × height (A = w × h). A square is a special type of rectangle where all sides are equal (w = h = s). Thus, the area of a square is side × side, or side squared (A = s²).
Can the area of a shape be an irrational number?
Yes, if the side lengths are irrational numbers (such as √2), the resulting area can be either rational or irrational. You can learn more about these classifications in our Number Sets - Real & Complex topic.