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Area: Square & Rectangle

Learn and practice calculating the area of squares, rectangles, composite shapes, and shapes with holes.

Square & Rectangle

Use the workspace below. Write equations like A = 30 to solve for the area.

Learning Topics

📖 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?

whRectangle

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.

ssssSquare

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.
561 × 1 unit square30 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 2\sqrt{2}). 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.

4. Area Formulas

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.
whA = w × h
Rectangle
ssA = s²
Square

Mastering SealMath: Entering the Square Root

To enter a square root (like 10\sqrt{10}) in the math input, you have two options:
• Keyboard shortcut: Type sqrt in the input box. MathLive will instantly create the root symbol □\sqrt{\square} 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).

How do you calculate the area of a right-angled triangle?

The area of a right-angled triangle is calculated by multiplying its two perpendicular legs and dividing by 2 (A = ab / 2). This is because a right-angled triangle is exactly half of a rectangle with the same width and height.

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.

Area: Square and Rectangle | SealMath