ASVAB geometry review
ASVAB geometry tests a short list of facts: angle sums, the Pythagorean theorem, similar triangles and the area and volume formulas for basic shapes. You do not need proofs. If you know the facts on this page and can apply them by hand, you can answer the geometry items on Mathematics Knowledge.
What geometry is on the ASVAB?
Geometry appears mainly on Mathematics Knowledge, which the official site describes as "knowledge of high school mathematics principles" (What to Expect). On the CAT-ASVAB, Mathematics Knowledge has 15 scored questions with a 31 minute limit when no tryout questions are added (Official ASVAB: CAT-ASVAB). The paper version has 25 MK questions in 24 minutes (What to Expect). Expect angles, triangles, circles, area, perimeter and volume. Arithmetic Reasoning also uses geometry inside word problems, such as how much carpet a room needs.
Here are the facts this page covers, in one place:
| Topic | Fact to know |
|---|---|
| Right angle | 90° |
| Straight line | 180°; two angles on a line are supplementary |
| Complementary angles | add to 90° |
| Vertical angles | opposite angles at an intersection are equal |
| Triangle | angles add to 180° |
| Quadrilateral | angles add to 360° |
| Polygon with n sides | angles add to (n − 2) × 180° |
| Isosceles triangle | two equal sides, two equal base angles |
| Equilateral triangle | three equal sides, each angle 60° |
| Similar figures | same shape, sides in the same ratio |
| Circle | radius is half the diameter; 360° around the center |
How do you find a missing angle?
Subtract the known angles from the total for that shape. Triangles total 180°, quadrilaterals 360°, and angles on a straight line 180°.
Example: Two angles of a triangle measure 47° and 68°. What is the third angle?
- Add the known angles: 47 + 68 = 115.
- Subtract from 180: 180 − 115 = 65.
Answer: 65°
Example: What is the sum of the interior angles of a hexagon? If the hexagon is regular, how big is each angle?
- A hexagon has 6 sides, so the sum is (6 − 2) × 180 = 4 × 180 = 720°.
- A regular hexagon has 6 equal angles: 720 ÷ 6 = 120°.
Answer: 720° in total, 120° each
Supplementary means the angles add to 180°. Complementary means they add to 90°. A memory hook: C comes before S, and 90 comes before 180.
How do you use the Pythagorean theorem in geometry questions?
For any right triangle, the two legs squared add up to the hypotenuse squared: a² + b² = c². Learn the common triples so you can often skip the squaring.
| Triple | Common multiples |
|---|---|
| 3-4-5 | 6-8-10, 9-12-15, 12-16-20 |
| 5-12-13 | 10-24-26 |
| 8-15-17 | 16-30-34 |
| 7-24-25 | 14-48-50 |
Example: A 13 foot ladder leans against a wall with its foot 5 feet from the wall. How high up the wall does it reach?
- The ladder is the hypotenuse: c = 13. One leg is 5.
- b² = 13² − 5² = 169 − 25 = 144.
- b = √144 = 12. This is the 5-12-13 triple.
Answer: 12 feet
How do you find the area of a combined shape?
Split it into shapes you know, find each area, and add (or subtract a cut-out). Write each piece separately so you do not lose one.
Example: A rug is a 10 by 6 foot rectangle with a half circle attached along one 6 foot end. Using π ≈ 3.14, what is its area?
- Rectangle: 10 × 6 = 60 square feet.
- Half circle: the diameter is 6, so r = 3. A full circle is π × 3² = 9π, half is 4.5π.
- 4.5 × 3.14 = 14.13.
- Total: 60 + 14.13 = 74.13.
Answer: about 74.13 square feet
Unit conversions inside area problems are a frequent trap. One square yard is 3 × 3 = 9 square feet, not 3 square feet.
Example: A room is 12 feet by 15 feet. How many square yards of carpet cover it?
- Area in square feet: 12 × 15 = 180.
- Divide by 9 square feet per square yard: 180 ÷ 9 = 20.
- Or convert first: 4 yards × 5 yards = 20.
Answer: 20 square yards
How do similar triangles work?
Similar triangles have equal angles and proportional sides. Set up a proportion with matching sides in matching positions. Shadow problems are the classic ASVAB version, because the sun hits the person and the object at the same angle.
Example: A 6 foot person casts a 4 foot shadow. At the same time a tree casts an 18 foot shadow. How tall is the tree?
- Height over shadow is the same for both: 6/4 = h/18.
- Cross multiply: 4h = 6 × 18 = 108.
- h = 108 ÷ 4 = 27.
Answer: 27 feet
Which area and volume formulas do you need for geometry?
You need the area of rectangles, triangles, parallelograms, trapezoids and circles, and the volume of boxes, cubes and cylinders. The full list with notes is on the ASVAB math formula sheet. A short version:
- Rectangle: l × w. Triangle: ½ × b × h. Trapezoid: ½ × (b₁ + b₂) × h.
- Circle: area πr², circumference 2πr.
- Box: l × w × h. Cylinder: πr²h.
If a question doubles every side of a square, the area grows four times (2 × 2). If it doubles every edge of a cube, the volume grows eight times (2 × 2 × 2). That idea answers "what happens to the area" questions without any numbers.
What angle rules apply to parallel lines?
When a straight line (a transversal) crosses two parallel lines, it creates only two angle sizes, and those two sizes add to 180°. Every acute angle in the picture is equal, and every obtuse angle is equal.
- Corresponding angles (same position at each crossing) are equal.
- Alternate interior angles (inside the parallel lines, on opposite sides of the transversal) are equal.
- Same-side interior angles (inside, on the same side) add to 180°.
- Vertical angles at any single crossing are equal.
Example: A transversal crosses two parallel lines and makes a 65° angle. What are the other angle sizes in the figure?
- Every angle equal to the 65° angle by the rules above is 65°.
- Every other angle is its supplement: 180 − 65 = 115°.
Answer: each angle is either 65° or 115°
On test day you rarely need the names. Decide whether the angle you want looks acute or obtuse, then use the given angle or 180 minus it.
How do you solve circle and coordinate geometry word problems?
Circle word problems usually hide the circumference: a wheel that turns once rolls forward one circumference. Coordinate questions usually ask for a distance or a midpoint between two points.
Example: A wheel has a diameter of 2 feet. About how far does it roll in 100 turns? Use π ≈ 3.14.
- One turn covers one circumference: π × d = 3.14 × 2 = 6.28 feet.
- 100 turns: 6.28 × 100 = 628 feet.
Answer: about 628 feet
Example: Find the distance and the midpoint between (1, 2) and (4, 6).
- Legs: 4 − 1 = 3 across and 6 − 2 = 4 up.
- Distance: √(3² + 4²) = √25 = 5 (a 3-4-5 triangle).
- Midpoint: average the coordinates. x: (1 + 4) ÷ 2 = 2.5. y: (2 + 6) ÷ 2 = 4.
Answer: distance 5, midpoint (2.5, 4)
A sketch on scratch work makes both types easier. Plot the two points roughly, draw the right triangle, and label its legs before you calculate. For circles, write r and d next to each other so you never square the wrong one.
How should you study geometry for Mathematics Knowledge?
Learn the facts table first, then practice drawing. A quick sketch with the given numbers written on it solves most geometry items faster than reading the text again.
- Copy the facts table and the triples from memory until you get them all right.
- Do the practice set below without looking at the formulas.
- Mix geometry with algebra on the Mathematics Knowledge practice page.
- For square roots and exponents that show up inside geometry, read exponents and roots.
Practice questions
Work each one on paper first, then open the explanation. These are original AptiStep items, not real ASVAB questions.
What is the supplement of a 38 degree angle?
- 52°
- 142°
- 322°
- 152°
Show answer and explanation
Answer: B. 142°
Supplementary angles add to 180°. 180 − 38 = 142°. The 52° choice is the complement (adding to 90°), which is the most common mix-up.
A right triangle has one acute angle of 35°. What is the other acute angle?
- 65°
- 145°
- 55°
- 45°
Show answer and explanation
Answer: C. 55°
The angles of a triangle add to 180°, and the right angle is 90°. 180 − 90 − 35 = 55°. A quicker route: the two acute angles of a right triangle add to 90°, so 90 − 35 = 55°.
What is the sum of the interior angles of a pentagon?
- 540°
- 360°
- 720°
- 900°
Show answer and explanation
Answer: A. 540°
Use (n − 2) × 180° with n = 5 sides: 3 × 180 = 540°. The 360° choice is the sum for a four-sided shape, and 720° is for a hexagon.
A rectangle has an area of 96 square feet and a length of 12 feet. What is its perimeter?
- 20 feet
- 48 feet
- 108 feet
- 40 feet
Show answer and explanation
Answer: D. 40 feet
Width = area ÷ length = 96 ÷ 12 = 8 feet. Perimeter = 2 × 12 + 2 × 8 = 24 + 16 = 40 feet. The 20 foot choice adds length and width only once.
The legs of a right triangle are 8 and 15. How long is the hypotenuse?
- 23
- 17
- 13
- 289
Show answer and explanation
Answer: B. 17
8² + 15² = 64 + 225 = 289. The square root of 289 is 17. This is the 8-15-17 triple. The 289 choice forgets the square root, and 23 just adds the legs.
A circle has a circumference of 18π. What is its area?
- 324π
- 18π
- 81π
- 36π
Show answer and explanation
Answer: C. 81π
Circumference = 2πr, so 2πr = 18π and r = 9. Area = πr² = π × 81 = 81π. The 324π choice squares the diameter (18) instead of the radius.
A 5 foot post casts a 3 foot shadow. At the same time a flagpole casts a 15 foot shadow. How tall is the flagpole?
- 25 feet
- 9 feet
- 17 feet
- 20 feet
Show answer and explanation
Answer: A. 25 feet
Height over shadow is equal for both: 5/3 = h/15. Cross multiply: 3h = 75, so h = 25 feet. The 9 foot choice flips the ratio (3/5 of 15).
A trapezoid has bases of 8 and 12 inches and a height of 5 inches. What is its area?
- 100 square inches
- 25 square inches
- 480 square inches
- 50 square inches
Show answer and explanation
Answer: D. 50 square inches
Area = ½ × (b₁ + b₂) × h = ½ × (8 + 12) × 5 = ½ × 20 × 5 = 50. The 100 choice forgets the one half, and 480 multiplies all three numbers.
Frequently asked questions
Do I need to know geometry proofs for the ASVAB?
No. ASVAB geometry asks you to use facts and formulas to find a number, such as an angle, a side, an area or a volume.
Is geometry on Arithmetic Reasoning too?
Yes, inside word problems: carpeting a room, fencing a yard or filling a tank. The math is the same; you just have to pull the shape out of the story.
What value of pi does the ASVAB use?
Use what the question gives. When choices are written with π, leave π in your answer. Otherwise 3.14 is the usual value.
Sources
Checked November 14, 2026. AptiStep is an independent study site; confirm anything that affects your enlistment with a recruiter.