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1. Word Knowledge

2. Mathematics Knowledge

2.1 Foundational math

2.2 Pre-algebra

2.3 Algebra I

2.4 Geometry basics

2.4.1 Lines and slopes

2.4.2 Angles and transversals

2.4.3 Basic shapes and area

2.4.4 Circles, polygons, and coordinate geometry

2.5 Algebra II

2.6 Advanced geometry

2.7 Math strategies

3. Paragraph Comprehension

4. Arithmetic Reasoning

5. General Science

6. Mechanical Comprehension

7. Electronics Information

8. Shop Information

9. Auto Information

10. Assembling Objects

Wrapping up

2.4.3 Basic shapes and area

Achievable ASVAB

2. Mathematics Knowledge

2.4. Geometry basics

Our ASVAB course is currently in development and is a work-in-progress.

Basic shapes and area

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Shapes

Definitions

Area: The space inside of an object
Perimeter: The distance measured around the outside of an object

When working with shapes, there are two main measurements you’ll see often: area and perimeter.

Area tells you how much space is inside a shape.
Perimeter tells you how far it is around the outside.

We’ll start with two of the most common shapes: rectangles and squares.

Rectangles and squares

Definitions

Rectangle: A polygon with exactly four sides that meet at four right angles ( $90$ °). The side lengths are called the length and the width.
Square: A special type of rectangle in which all four side lengths are equal

A rectangle has four sides and four right angles ( $9 0^{\circ}$ ). We call the side lengths the length and the width.

The interior angles of a rectangle always add to $36 0^{\circ}$ because:

$9 0^{\circ} \times 4 = 36 0^{\circ}$

A square is a special type of rectangle where all four sides are the same length.

A square with all four sides marked equal and a right angle at each corner. — Square

Finding area and perimeter

To find the area, multiply the length by the width:

$A re a = l e n g t h \times w i d t h$

To find the perimeter, add all the sides together:

$P er im e t er = l e n g t h + l e n g t h + w i d t h + w i d t h$

This can also be written more simply as:

$P er im e t er = 2 (l e n g t h) + 2 (w i d t h)$

Example:

A rectangle has a perimeter of $36$ inches. If the rectangle has a length of $12$ inches, what is the width of the rectangle?

This is asking: what value makes the perimeter formula equal to $36$ ?

Use the perimeter formula:

$P er im e t er = 2 (l e n g t h) + 2 (w i d t h)$

Substitute the known values:

$3636 = 2 (12) + 2 (w i d t h) = 24 + 2 w$

Now solve:

$126 = 2 w = w$

The width of the rectangle is $6$ inches.

Steps taken:

Use the perimeter formula
Substitute known values
Solve for the unknown
Write the final answer

Knowledge check: The area of a square is $64 in^{2}$ . What is the side length of the square?

(spoiler)

$8$ inches.

To find the side length, take the square root of the area:

$64 = 8$

Triangles

Definitions

Triangle: A three-sided polygon whose interior angles always sum to $180$ °

The interior angles of a triangle always add up to $18 0^{\circ}$ . This is important because if you know two angles, you can always find the third. One way to remember this is to picture a triangle as half of a rectangle. Since a rectangle’s interior angles sum to $36 0^{\circ}$ , half of that is:

$\frac{360}{2} = 18 0^{\circ}$

A rectangle cut along its diagonal into two identical right triangles. — Rectangle to triangle

Types of triangles

Definitions

Isosceles triangle: A triangle with two congruent sides and two congruent angles.
Equilateral triangle: A triangle in which all sides and angles are equal.
Right triangle: A triangle that has one right angle ( $90$ °)
Obtuse triangle: A triangle that has one angle larger than $90$ °
Acute triangle: A triangle that has all three angles less than $90$ °

There are many different types of triangles! Triangles can be grouped based on their side lengths or their angle sizes. Knowing the type of triangle helps you quickly understand its properties and solve problems more easily.

Some triangles are classified by their sides (like isosceles and equilateral).
Others are classified by their angles (like right, obtuse, and acute).

For example, a right triangle will always have one $9 0^{\circ}$ angle, while an equilateral triangle will always have all sides and angles equal.

Five triangles: an isosceles triangle with two equal sides and two equal base angles, an equilateral triangle with three equal sides and angles, a right triangle with one 90° angle, an obtuse triangle with one angle over 90°, and an acute triangle with every angle under 90°. — Types of triangles

Area of triangles

Definitions

Altitude: A perpendicular line drawn from a vertex of a triangle to the opposite side. This line represents the height in the area formula.

To find the area of a triangle, you need two things:

The base (a side of the triangle), and
The height (how tall the triangle is).

The height is not always one of the triangle’s sides. Instead, it is the perpendicular distance from the base to the opposite vertex.

The formula for the area of a triangle is:

$A re a = \frac{1}{2} \times ba se \times h e i g h t$

This formula comes from the idea that a triangle is half of a rectangle. If you imagine two identical triangles put together, they form a rectangle.

$A re a = \frac{1}{2} \times l e n g t h \times w i d t h$

Sometimes it’s easy to see the base and height, but other times you need to look more carefully.

For right triangles, the base and height are already given by the sides that form the right angle, so they are easy to identify.

A right triangle. The two sides that form the right angle are labeled height and base. — Right triangle

For many other triangles, you’ll need to draw the height down to the base. This height line is also called an altitude.

The altitude of a triangle is a perpendicular line drawn from a vertex to the opposite side. Sometimes you may need to extend the base to draw the altitude.

Two triangles with their altitudes drawn as dashed lines at right angles to the base. In the first the altitude falls inside the triangle; in the second it falls outside, onto the base extended. — Altitude of a triangle

Once you have the altitude, use it as the height in the area formula.

Now let’s apply what we’ve learned about base and height to find the area of a triangle.

Example:

What is the area of the triangle?

This is asking: what is the space inside the triangle?
$A triangle with a base of 4 cm and a dashed altitude of 6 cm.$
Triangle example problem
Achievable

Use the altitude as the height. From the figure:

The height is $6$ cm

The base is $4$ cm

Now substitute into the formula:

$A re a A re a A = \frac{1}{2} \times ba se \times h e i g h t = \frac{1}{2} \times 4 \times 6 = 12 cm^{2}$

The area of the triangle is $12 cm^{2}$ .

Steps taken:

Identify the base and height
Use the altitude as the height when needed
Substitute into the area formula
Multiply to find the area

Knowledge check: What is the area of a triangle with a base of $10$ cm and a height of $5$ cm?

(spoiler)

Use the formula:

$A re a = \frac{1}{2} \times ba se \times h e i g h t$

$\frac{1}{2} \times 10 \times 5 = 25$

The area is $25 cm^{2}$ .

Area & Perimeter Basics

Area: space inside a shape
Perimeter: distance around outside of a shape

Rectangles and Squares

Rectangle: 4 sides, 4 right angles ( $90°$ each), interior angles sum to $360°$
Square: special rectangle with all 4 sides equal
Formulas:
- $A re a = l e n g t h \times w i d t h$
- $P er im e t er = 2 (l e n g t h) + 2 (w i d t h)$
To solve for unknown side: plug known values into formula, solve algebraically
Square side length = $A re a$

Triangles: Angle Sum & Types

Interior angles of any triangle always sum to $180°$ (half of a rectangle’s $360°$ )
Knowing two angles lets you find the third
Classified by sides: isosceles (2 equal sides/angles), equilateral (all equal)
Classified by angles: right ( $90°$ ), obtuse (> $90°$ ), acute (all < $90°$ )

Area of Triangles

Formula: $A re a = \frac{1}{2} \times ba se \times h e i g h t$
Derived from triangle being half of a rectangle
Height = perpendicular distance from base to opposite vertex, not always a side
Altitude: the perpendicular line representing height; may require extending the base to draw it
For right triangles, the two legs forming the right angle serve directly as base and height

Next | 2.4.4 Circles, polygons, and coordinate geometry

Basic shapes and area

Shapes

Definitions

Area: The space inside of an object
Perimeter: The distance measured around the outside of an object

When working with shapes, there are two main measurements you’ll see often: area and perimeter.

Area tells you how much space is inside a shape.
Perimeter tells you how far it is around the outside.

We’ll start with two of the most common shapes: rectangles and squares.