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.

2. Right-angled Triangle

Learn how a right-angled triangle forms half of a rectangle and practice calculating its area.

3. Triangle

Learn how to calculate the area of any triangle using the base and height, and practice dynamic exercises.

4. Circle

Learn and practice calculating the area of circles, sectors, composite shapes, and shapes with cutouts.

5. Dimensions & Units

Understand dimensions and units of area. Convert between metric (mm2, cm2, m2) and US customary (in2, ft2, yd2) area units, and practise ratio problems with composite shapes.

Learning Guide

1. Square & Rectangle

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).

2. Right-angled Triangle

1. The Right-angled Triangle

Right-Angled Triangle

A right-angled triangle is a three-sided flat shape that has one interior angle of exactly

90^\circ

Completing to a Rectangle & Congruence

Completing to a Rectangle

If we take any right-angled triangle with legs a and b and copy it, we can flip the copy and place it next to the original to form a complete rectangle with side lengths a and b.

Definition: Congruent Triangles

Triangles are congruent when they are identical in shape and size, meaning all of their corresponding sides and angles match perfectly.

Since our original triangle and its flipped copy share these identical properties, these two triangles are congruent.

We can show this here using the Z-rule (alternate interior angles) we learned earlier:
• If the bottom angle of our original triangle is

\alpha

and the top angle is

\beta

(where

\alpha + \beta = 90^\circ

), the Z-rule shows the opposite angles in the completed triangle must also be

\alpha

and

\beta

.
• Since both triangles together make a rectangle with all angles 90°, their opposite sides are equal, meaning the sides of both triangles are identical.

Because both triangles are congruent, they cover the exact same amount of space and have the same area. Therefore, the area of the right-angled triangle is exactly half of the rectangle's area:

A = \frac{a \times b}{2} = \frac{ab}{2}

2. Area Formulas

Right-angled Triangle

Area of a Right-angled Triangle

A = \frac{ab}{2}

3. Triangle

1. Base and Height of a Triangle

To calculate the area of any triangle, we need two key measurements:
• Base (b): Any of the triangle's three sides.
• Height (h): The perpendicular line segment from the opposite vertex to that base (or its extension). The height is always at a 90° angle to the base.

Height of a Triangle

The height of a triangle relative to a base is the perpendicular distance from that base (or its extension) to the opposite vertex.

Acute Triangle

An acute triangle is a triangle where all three interior angles are acute (each less than

90^\circ

Acute Triangle (Height Inside)

In an acute triangle (where all angles are less than 90°), the height to the base falls inside the triangle.

Obtuse Triangle

An obtuse triangle is a triangle that has one interior angle that is obtuse (greater than

90^\circ

Obtuse Triangle (Height Outside)

In an obtuse triangle (where one angle is greater than 90°), the height to the base can fall outside the triangle. To measure it, we extend the base line outwards.

2. Intersection of Heights (Orthocenter)

Every triangle has three heights (one for each of its three sides). An elegant property of geometry is that all three heights (or their extensions) intersect at exactly one point. This intersection point is called the orthocenter.

3. Triangle Area Formula

Since any triangle is exactly half of a parallelogram (or rectangle) with the same base and height, its area is given by multiplying the base and height and dividing by 2:

A = \frac{b \times h}{2}

Where:
•

b

is the length of the base (opposite side).
•

h

is the height perpendicular to that base.

Area of a Triangle

A = \frac{b \times h}{2}

4. Circle

1. What Are a Circle, Radius, and Diameter?

Circle

A circle is a round flat shape where all points on the boundary are at the same distance from the center.

Circle: Radius & Diameter

2. Circle Area Formula

The area of a full circle is calculated using its radius and the constant

\pi

(pi, which is approximately

3.14159

). Note that

\pi

is an irrational number (

\pi \in \mathbb{R}

and

\pi \notin \mathbb{Q}

), meaning it cannot be written as a simple fraction and has infinite non-repeating decimal digits. To learn more about this classification, refer to the Real & Complex Number Sets topic.

Area of a Circle

A = \pi r^2

3. Area of a Sector

Circle Sector

A circle sector is the portion of a disk enclosed by two radii and an arc.

• In Radians (central angle

\theta

Area of a Circle Sector (Radians)

A = \frac{1}{2} r^2 \theta

Calculated using central angle

\theta

in radians.

• In Degrees (central angle

\theta

Area of a Circle Sector (Degrees)

A = \frac{\theta}{360^\circ} \times \pi r^2

Calculated using central angle

\theta

in degrees.

• Common angles:

180^\circ

(semicircle, half of a circle) and

90^\circ

(quadrant, quarter of a circle).

4. Deriving the Angle from Area

If we are given the area

A

of a sector and its radius

r

, we can work backward to find the angle

\theta

by using the derived formulas:

• In Radians:

Angle of a Circle Sector (Radians)

\theta = \frac{2A}{r^2}

Calculated using area A and radius r in radians.

• In Degrees:

Angle of a Circle Sector (Degrees)

\theta = \frac{360^\circ \times A}{\pi r^2}

Calculated using area A and radius r in degrees.

Mastering SealMath: Entering Pi ( $\pi$ )

To enter the pi symbol (

\pi

) in the math input, you have three options:
• Keyboard shortcut: Type pi in the input box. It will instantly convert to

\pi

.
• LaTeX input: Type \pi and press Enter (or click on the popup suggestion) to get the symbol

\pi

.
• Virtual keyboard: Click the ⌨️ keyboard icon inside the input box to open the on-screen keyboard, switch to the greek tab, and press the

\pi

button.

5. Dimensions & Units

1. What Is a Dimension?

A dimension tells us how many independent directions we need to measure a shape.
• A line is 1-dimensional (1D) — we only need one measurement: its length.
• A flat surface (like a rectangle or circle) is 2-dimensional (2D) — we need two measurements: width and height. This is why area is measured in square units.
• A solid object (like a box) is 3-dimensional (3D).

2. US Customary Units of Length

The US customary system is used primarily in the United States. Unlike the metric system, it does not follow powers of 10:

Units (small → large):
• in (inch): about the width of a thumb — 12 inches = 1 foot
• ft (foot): about the height of a standard ceiling tile — 1 ft = 12 in
• yd (yard): about one long step — 1 yd = 3 ft = 36 in
• mi (mile): about 20 minutes walking — 1 mi = 1,760 yd = 5,280 ft

US Customary Length Conversions

1\text{ mi} = 1{,}760\text{ yd} \\ 1\text{ yd} = 3\text{ ft} \\ 1\text{ ft} = 12\text{ in}

3. Metric Units of Length

The metric system is used in most countries worldwide. It is based entirely on powers of 10:

Units (small → large):
• mm (millimetre): about the width of a pencil tip
• cm (centimetre): about the width of a fingernail — 1 cm = 10 mm
• dm (decimetre): 1 dm = 10 cm = 100 mm
• m (metre): about the height of a door — 1 m = 100 cm = 1,000 mm
• km (kilometre): about 10 minutes walking — 1 km = 1,000 m

Metric Length Conversions

1\text{ m} = 10\text{ dm} \\ 1\text{ dm} = 10\text{ cm} \\ 1\text{ cm} = 10\text{ mm}

4. Converting Between Length Units

To convert a length, multiply when going from a larger unit to a smaller one, and divide when going from a smaller unit to a larger one:

US Customary:
• ft → in: multiply by 12 (e.g. 2 ft = 24 in)
• in → ft: divide by 12
• yd → ft: multiply by 3
• ft → yd: divide by 3

Metric:
• cm → mm: multiply by 10 (e.g. 5 cm = 50 mm)
• mm → cm: divide by 10 (e.g. 30 mm = 3 cm)
• m → cm: multiply by 100
• cm → m: divide by 100

Metric Prefixes

\begin{aligned} \text{tera}\text{ (T)} &= 10^{12} \\ \text{giga}\text{ (G)} &= 10^9 \\ \text{mega}\text{ (M)} &= 10^6 \\ \text{kilo}\text{ (k)} &= 10^3 \\ \text{centi}\text{ (c)} &= 10^{-2} \\ \text{milli}\text{ (m)} &= 10^{-3} \\ \text{micro}\text{ (}\mu\text{)} &= 10^{-6} \\ \text{nano}\text{ (n)} &= 10^{-9} \\ \text{pico}\text{ (p)} &= 10^{-12} \end{aligned}

Scale factors relative to the base unit

5. Converting Between US & Metric Units

Sometimes we need to convert measurements between the US customary and metric systems. Here are the most common length conversion factors:

• inch to cm: multiply by 2.54 (e.g., 2 in = 5.08 cm)
• foot to cm: multiply by 30.48
• mile to km: multiply by 1.609 (e.g., 10 mi ≈ 16.09 km)

To go the opposite way (metric to US customary), simply divide by these conversion factors instead of multiplying.

US & Metric Length Conversions

\begin{aligned} 1\text{ in} &= 2.54\text{ cm} \\ 1\text{ ft} &= 30.48\text{ cm} \\ 1\text{ mi} &\approx 1.609\text{ km} \end{aligned}

Converting between US Customary and Metric length units

6. Units of Area

Because area is length × length, the unit of area is the square of the length unit:

Metric area units: mm², cm², dm², m², km²
US area units: in², ft², yd², mi²

For example, 1 cm² means one square that is 1 cm wide and 1 cm tall. A rectangle 3 cm wide and 5 cm tall has an area of 15 cm².

7. Converting Area Units — The Critical Insight

⚠️ Area conversion is NOT the same as length conversion!

Because area = length × length, when you convert the length unit, you must apply the conversion factor twice (once for each dimension).

Example: 1 cm² to mm²
1 cm = 10 mm, so:
1 cm² = 1 cm × 1 cm = 10 mm × 10 mm = 100 mm²

1 cm² = 100 mm² (not 10!)

Think of it this way: a 1 cm × 1 cm square can be divided into a 10 × 10 grid of 1 mm × 1 mm squares — that is 100 tiny squares in total.

Area Unit Conversion Rule

\text{if}\ 1\ \text{unit} = k\ \text{sub-units} \\ \text{then}\ 1\ \text{unit}^2 = k^2\ \text{sub-units}^2

Metric Area Conversions

1\text{ m}^2 = 100\text{ dm}^2 \\ 1\text{ dm}^2 = 100\text{ cm}^2 \\ 1\text{ cm}^2 = 100\text{ mm}^2

US Customary Area Conversions

1\text{ mi}^2 = 3{,}097{,}600\text{ yd}^2 \\ 1\text{ yd}^2 = 9\text{ ft}^2 \\ 1\text{ ft}^2 = 144\text{ in}^2

8. Ratio of Areas — Same Dimensions & Units

A ratio compares two quantities of the same type and same unit. When both areas share the same unit, the ratio is a pure number — the units cancel out.

Example: Part A has an area of 12 cm². Part B has an area of 8 cm².
Ratio of A to B = 12 cm² / 8 cm² = 12/8 = 3/2
Written as: 3:2 (simplified).

Important: If the two areas are in different units, you must convert them to the same unit first, then take the ratio.

Enter your ratio answer using the a:b format (e.g. 3:2), fully simplified.

Mastering SealMath: Ratio & Unit Answers

For ratio problems: enter your answer in a:b format — for example, type 3:2 for a ratio of 3 to 2. Make sure the ratio is fully simplified (no common factors).

For area calculation problems: enter your numerical answer with A = value as usual. The unit is displayed in the question — you do not need to type it in the answer.

Learning Topics

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

Area Study Guide

Area Practice Topics

1. Square & Rectangle

2. Right-angled Triangle

3. Triangle

4. Circle

5. Dimensions & Units

Learning Guide

1. Square & Rectangle

1. What Are a Rectangle and a Square?

Rectangle

Square

2. Area as a 1 × 1 Grid

3. Rational and Irrational Area

4. Area Formulas

Area of a Rectangle

Area of a Square

5. Composite Areas & Cutouts

Area of Composite Shapes

Area of Cutouts & Holes

Mastering SealMath: Entering the Square Root

2. Right-angled Triangle

1. The Right-angled Triangle

Right-Angled Triangle

Completing to a Rectangle & Congruence

Definition: Congruent Triangles

2. Area Formulas

Area of a Right-angled Triangle

3. Triangle

1. Base and Height of a Triangle

Height of a Triangle

Acute Triangle

Obtuse Triangle

2. Intersection of Heights (Orthocenter)

3. Triangle Area Formula

Area of a Triangle

4. Circle

1. What Are a Circle, Radius, and Diameter?

Circle

2. Circle Area Formula

Area of a Circle

3. Area of a Sector

Circle Sector

Area of a Circle Sector (Radians)

Area of a Circle Sector (Degrees)

4. Deriving the Angle from Area

Angle of a Circle Sector (Radians)

Angle of a Circle Sector (Degrees)

Mastering SealMath: Entering Pi (π\piπ)

5. Dimensions & Units

1. What Is a Dimension?

2. US Customary Units of Length

US Customary Length Conversions

3. Metric Units of Length

Metric Length Conversions

4. Converting Between Length Units

Metric Prefixes

5. Converting Between US & Metric Units

US & Metric Length Conversions

6. Units of Area

7. Converting Area Units — The Critical Insight

Area Unit Conversion Rule

Metric Area Conversions

US Customary Area Conversions

8. Ratio of Areas — Same Dimensions & Units

Mastering SealMath: Ratio & Unit Answers

Frequently Asked Questions

Mastering SealMath: Entering Pi ( $\pi$ )