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.

How do you calculate the area of a general triangle?

The area of any general triangle is calculated as half the base times the height (A = (1/2) × b × h or A = bh/2), where the height is the perpendicular distance from the base to the opposite vertex.

How do you calculate the area of a circle?

The area of a circle is calculated as π times the radius squared (A = πr²). Since π (pi) is an irrational number, the area of a circle with a rational radius will always be an irrational number.

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). It is used to find a missing side length when the other two are known.

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.

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.

What are the main types of special triangles?

The three main types are: isosceles (two equal sides and two equal base angles), equilateral (all sides and angles equal — each angle is 60°), and the 30-60-90 right triangle (fixed side ratios of 1 : √3 : 2).

Why does 1 cm² equal 100 mm² and not just 10 mm²?

Because area is two-dimensional (length × length), the conversion factor must be applied twice. Since 1 cm = 10 mm, we have 1 cm² = 1 cm × 1 cm = 10 mm × 10 mm = 100 mm². In general, if 1 unit = k sub-units, then 1 unit² = k² sub-units². The same logic applies to all other area unit conversions.

Area: Square and Rectangle | SealMath