Textbook

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

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

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

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

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

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:

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

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:

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

For a square with side $s$ , the length and width are equal, so the formulas simplify to $A r e a = s \times s = s^{2}$ and $P er im e t er = 4 s$ . To go from area back to the side length, take the square root - for example, $64 = 8$ . Memorizing perfect squares up to $1 5^{2} = 225$ makes these easy to spot by hand, since the ASVAB math test doesn’t allow a calculator.

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.

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

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 cut a rectangle along its diagonal into two right triangles; each gets half of the rectangle’s $36 0^{\circ}$ , or $18 0^{\circ}$ . The $18 0^{\circ}$ rule holds for every triangle, not just right ones.

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

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

Types of triangles

Definitions

Isosceles triangle: A triangle with at least two congruent sides (so an equilateral triangle counts too); the angles opposite those sides are congruent.
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 $90$ °

Isosceles and equilateral describe a triangle’s sides; right, obtuse, and acute describe its angles.

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:

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

Two identical copies of any triangle fit together into a parallelogram (a rectangle, if the triangle is a right triangle) with the same base and height, so the triangle has half that area.

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. 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.

Watch out: the height must be perpendicular to the base - a true right angle to the base, not a slanted side of the triangle. Using a slanted side as the height gives you the wrong area.

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:

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

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

Knowledge check: A triangle has angles of $5 0^{\circ}$ and $6 0^{\circ}$ . What is the measure of the third angle?

(spoiler)

$7 0^{\circ}$ .

The three angles of a triangle always add up to $18 0^{\circ}$ :

$180 - 50 - 60 = 70$

Rectangles and Squares

Rectangle: four sides, four $90°$ angles; angles sum to $360°$
Square: special rectangle with all sides equal
Area $= l e n g t h \times w i d t h$ ; Perimeter $= 2 (l e n g t h) + 2 (w i d t h)$
Square simplifies: $A r e a = s^{2}$ , $P er im e t er = 4 s$ ; reverse area to side via square root
Memorize perfect squares up to $1 5^{2} = 225$ (no calculator on ASVAB)

Triangles: Basics and Types

Triangle angles always sum to $180°$ (half of a rectangle’s $360°$ )
Side-based types: isosceles (≥2 congruent sides, incl. equilateral), equilateral (all sides/angles equal)
Angle-based types: right ( $90°$ angle), obtuse (> $90°$ angle), acute (all angles < $90°$ )
Knowing two angles lets you find the third by subtracting from $180°$

Area of Triangles

Formula: $A r e a = \frac{1}{2} \times ba se \times h e i g h t$
Height = altitude, the perpendicular distance from base to opposite vertex (not necessarily a side)
Right triangles: legs forming the right angle serve directly as base/height
Other triangles: may need to draw/extend altitude to the base
Caution: height must be perpendicular to base — using a slanted side gives wrong area

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

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