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?

Square

A square is a four-sided flat shape with four equal sides and four right angles (

90^\circ

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

\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.

Rectangle

Square

Area of a Rectangle

A = w \times h

Area of a Square

A = s^2

5. Composite Areas & Cutouts

To calculate the area of more complex shapes, we use the properties of addition and subtraction:

Area of Composite Shapes

A_{\text{total}} = A_1 + A_2 + \dots

The total area is the sum of the areas of the individual shapes.

For shapes with hollow spaces, holes, or cutouts, we subtract the areas:

Area of Cutouts & Holes

A_{\text{remaining}} = A_{\text{outer}} - A_{\text{inner}}

The remaining area is the outer area minus the inner cutout area.

Mastering SealMath: Entering the Square Root

To enter a square root (like

\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 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 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.

What is the area ratio of inscribed circles and squares?

For a square inscribed in a circle, the area ratio of the square to the circle is always exactly 2/π ≈ 0.637. For a circle inscribed in a square, the area ratio of the circle to the square is always exactly π/4 ≈ 0.785. These ratios are constant regardless of the shapes' actual sizes.