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Math Knowledge

Geometry & Measurement

Geometry focuses on shapes, distances, angles, area, and space. For ASVAB Math Knowledge, the most important geometry skills are knowing the basic formulas and recognizing which formula a problem is asking you to use. Many geometry problems become simple once you identify what information you are given, what measurement the question wants, and which formula connects those values. Pay close attention to units: length uses regular units, area uses square units, and volume uses cubic units.

Section 1

Perimeter

Perimeter is the total distance around the outside of a two-dimensional shape. Think of it as the material needed to form a border, such as fencing around a yard, trim around a room, or the outside edge of a sign.

Rectangle Perimeter

Use P=2l+2w, where l is length and w is width. You can also write P=2(l+w).

Square Perimeter

All four sides of a square are equal. Use P=4s, where s is the side length.

Triangle Perimeter

Add all three side lengths: P=a+b+c.

Rules and formulas

  • Rectangle: P=2l+2w.
  • Square: P=4s.
  • Triangle: add all three sides.

Example 1

A rectangle is 9 feet long and 5 feet wide. Find its perimeter.

Solution

  1. 1Use P=2l+2w.
  2. 2Substitute: P=2(9)+2(5).
  3. 3Multiply: P=18+10.
  4. 4Add: P=28.
Answer: 28 feet

Why: Perimeter measures the entire distance around the rectangle, so both lengths and both widths must be included.

Example 2

A square has sides that are 7 inches long. Find its perimeter.

Solution

  1. 1Use P=4s.
  2. 2Substitute: P=4(7).
  3. 3Multiply: P=28.
Answer: 28 inches

Why: A square has four equal sides, so multiplying one side by 4 gives the total distance around it.

Example 3

A triangular sign has side lengths of 8 inches, 11 inches, and 13 inches. Find its perimeter.

Solution

  1. 1Add all three sides.
  2. 2P=8+11+13.
  3. 3P=32.
Answer: 32 inches

Why: The perimeter of any polygon is found by adding the lengths of all its outside sides.

Section 2

Area

Area measures the amount of surface inside a two-dimensional shape. Area is measured in square units such as square feet, square inches, or square meters.

Rectangle Area

Use A=lw, where l is length and w is width.

Square Area

Because all sides are equal, use A=s2.

Triangle Area

Use A=12bh, where b is the base and h is the perpendicular height.

The height must be measured straight from the base at a right angle.

Circle Area

Use A=π r2, where r is the radius.

Rules and formulas

  • Rectangle: A=lw.
  • Square: A=s2.
  • Triangle: A=12bh.
  • Circle: A=π r2.

Example 1

A rectangular floor is 12 feet long and 8 feet wide. Find its area.

Solution

  1. 1Use A=lw.
  2. 2Substitute: A=12(8).
  3. 3Multiply: A=96.
Answer: 96 ft2

Why: Area measures the surface inside the rectangle, so its length is multiplied by its width.

Example 2

A triangle has a base of 10 inches and a height of 7 inches. Find its area.

Solution

  1. 1Use A=12bh.
  2. 2Substitute: A=12(10)(7).
  3. 3Multiply: 10×7=70.
  4. 4Take half: A=35.
Answer: 35 in2

Why: A triangle with the same base and height as a rectangle takes up half of that rectangle's area.

Example 3

A square has sides measuring 9 meters. Find its area.

Solution

  1. 1Use A=s2.
  2. 2Substitute: A=92.
  3. 3Evaluate: A=81.
Answer: 81 m2

Why: A square's length and width are equal, so multiplying the side by itself gives the area.

Section 3

Angles

An angle measures the amount of rotation between two lines or rays. Angles are measured in degrees, shown with the symbol °.

Types of Angles

An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is greater than 90° but less than 180°. A straight angle is exactly 180°.

Complementary Angles

Two angles are complementary when their measures add to 90°. For example, 30°+60°=90°.

Supplementary Angles

Two angles are supplementary when their measures add to 180°. For example, 110°+70°=180°.

Vertical Angles

When two lines cross, opposite angles are vertical angles. Vertical angles are equal.

Angle Sums

The three interior angles of a triangle add to 180°. The four interior angles of a quadrilateral add to 360°.

Rules and formulas

  • Complementary angles sum to 90°.
  • Supplementary angles sum to 180°.
  • Triangle angles sum to 180°.
  • Quadrilateral angles sum to 360°.
  • Vertical angles are equal.

Example 1

Two angles are complementary. One measures 34°. Find the other angle.

Solution

  1. 1Complementary angles total 90°.
  2. 2Subtract: 90°-34°=56°.
Answer: 56°

Why: Complementary angles must add to exactly 90°.

Example 2

A triangle has angles measuring 48°, 67°, and x°. Find x.

Solution

  1. 1Triangle angles total 180°.
  2. 2Add the known angles: 48°+67°=115°.
  3. 3Subtract from 180°: x=180°-115°.
  4. 4x=65°.
Answer: 65°

Why: The three interior angles of every triangle always add to 180°.

Example 3

Two lines intersect. One angle measures 118°. Find the adjacent angle that forms a straight line with it.

Solution

  1. 1Angles forming a straight line are supplementary.
  2. 2They total 180°.
  3. 3Subtract: 180°-118°=62°.
Answer: 62°

Why: Adjacent angles along a straight line must add to 180°.

Section 4

Pythagorean Theorem

The Pythagorean Theorem applies to right triangles, which contain one 90° angle. Use a2+b2=c2, where a and b are the legs and c is the hypotenuse. The hypotenuse is opposite the right angle and is the longest side.

Finding the Hypotenuse

Square each leg, add the results, and take the square root.

Finding a Missing Leg

Square the hypotenuse, subtract the square of the known leg, and take the square root.

Common Pythagorean Triples

Recognizing 3-4-5, 5-12-13, and 8-15-17 can save time. Multiples such as 6-8-10 also work.

Rules and formulas

  • Use only for right triangles.
  • a2+b2=c2.
  • c is the hypotenuse and longest side.

Example 1

A right triangle has legs of 6 and 8. Find the hypotenuse.

Solution

  1. 1Use a2+b2=c2.
  2. 2Substitute: 62+82=c2.
  3. 3Square: 36+64=c2.
  4. 4Add: 100=c2.
  5. 5Take the square root: c=√100=10.
Answer: 10

Why: The Pythagorean Theorem connects the two legs of a right triangle to its hypotenuse.

Example 2

A right triangle has hypotenuse 13 and one leg 5. Find the missing leg.

Solution

  1. 1Let the missing leg be b and use a2+b2=c2.
  2. 2Substitute: 52+b2=132.
  3. 3Square: 25+b2=169.
  4. 4Subtract 25: b2=144.
  5. 5Take the square root: b=√144=12.
Answer: 12

Why: When the hypotenuse is known, subtract the square of the known leg from the square of the hypotenuse.

Example 3

A ladder reaches 15 feet up a wall, and its bottom is 8 feet from the wall. How long is the ladder?

Solution

  1. 1The wall and ground form a right angle, and the ladder is the hypotenuse.
  2. 2Use a2+b2=c2.
  3. 3Substitute: 152+82=c2.
  4. 4Square and add: 225+64=289=c2.
  5. 5Take the square root: c=√289=17.
Answer: 17 feet

Why: The ladder is opposite the right angle, so it is the hypotenuse of the right triangle.

Section 5

Circles

A circle has several important measurements. The radius runs from the center to the edge. The diameter crosses the entire circle through its center and is twice the radius.

Radius and Diameter

Use d=2r and r=d2.

Circumference

Circumference is the distance around a circle. Use C=2π r or C=π d.

Area of a Circle

Use A=π r2. Circumference is a length, while area uses square units.

Rules and formulas

  • Diameter: d=2r.
  • Radius: r=d2.
  • Circumference: C=2π r or C=π d.
  • Area: A=π r2.

Example 1

A circle has a radius of 6 cm. Find its circumference.

Solution

  1. 1Use C=2π r.
  2. 2Substitute: C=2π(6).
  3. 3Multiply: C=12π.
Answer: 12π cm

Why: Circumference measures the distance around the circle, and the formula uses twice the radius multiplied by π.

Example 2

A circle has a diameter of 14 inches. Find its circumference.

Solution

  1. 1Since the diameter is given, use C=π d.
  2. 2Substitute: C=π(14).
Answer: 14π inches

Why: When the diameter is already known, C=π d is the fastest circumference formula.

Example 3

A circle has a radius of 5 feet. Find its area.

Solution

  1. 1Use A=π r2.
  2. 2Substitute: A=π(52).
  3. 3Square the radius: 52=25.
  4. 4Therefore, A=25π.
Answer: 25π ft2

Why: Area measures the space inside the circle, so the radius is squared.

Section 6

Volume

Volume measures the amount of space inside a three-dimensional object. It is measured in cubic units such as ft3, in3, or m3.

Rectangular Prism

Use V=lwh, where l is length, w is width, and h is height.

Cube

All sides have the same length, so use V=s3.

Cylinder

Use V=π r2h. A cylinder's volume is the area of its circular base multiplied by its height.

Rules and formulas

  • Rectangular prism: V=lwh.
  • Cube: V=s3.
  • Cylinder: V=π r2h.

Example 1

A rectangular box is 8 inches long, 5 inches wide, and 3 inches high. Find its volume.

Solution

  1. 1Use V=lwh.
  2. 2Substitute: V=8(5)(3).
  3. 3Multiply: 8×5=40, then 40×3=120.
Answer: 120 in3

Why: Volume measures three-dimensional space, so length, width, and height are all multiplied together.

Example 2

A cube has side length 4 feet. Find its volume.

Solution

  1. 1Use V=s3.
  2. 2Substitute: V=43.
  3. 3Calculate: 4×4×4=64.
Answer: 64 ft3

Why: A cube has equal length, width, and height, so multiplying the same side three times gives its volume.

Example 3

A cylinder has radius 3 cm and height 8 cm. Find its volume.

Solution

  1. 1Use V=π r2h.
  2. 2Substitute: V=π(32)(8).
  3. 3Square the radius: 32=9.
  4. 4Multiply: V=π(9)(8)=72π.
Answer: 72π cm3

Why: The cylinder's volume equals the area of its circular base multiplied by its height.

Section 7

Surface Area

Surface area measures the total area covering the outside of a three-dimensional object. Think of it as the material needed to completely cover the outside. Surface area uses square units. Volume measures inside space; surface area measures outside covering.

Cube Surface Area

A cube has six identical square faces. Use SA=6s2.

Rectangular Prism Surface Area

A rectangular prism has three pairs of equal faces. Use SA=2lw+2lh+2wh.

Rules and formulas

  • Cube: SA=6s2.
  • Rectangular prism: SA=2lw+2lh+2wh.

Example 1

A cube has a side length of 5 inches. Find its surface area.

Solution

  1. 1Use SA=6s2.
  2. 2Substitute: SA=6(52).
  3. 3Square the side: 52=25.
  4. 4Multiply: SA=6(25)=150.
Answer: 150 in2

Why: A cube contains six identical square faces, each with area s2.

Example 2

A rectangular prism has length 6 feet, width 4 feet, and height 3 feet. Find its surface area.

Solution

  1. 1Use SA=2lw+2lh+2wh.
  2. 2Substitute: SA=2(6)(4)+2(6)(3)+2(4)(3).
  3. 3Calculate: 48+36+24.
  4. 4Add: SA=108.
Answer: 108 ft2

Why: A rectangular prism has two faces for each length-width, length-height, and width-height combination.

Example 3

A cube has a surface area of 216 cm2. Find its side length.

Solution

  1. 1Use SA=6s2.
  2. 2Substitute: 216=6s2.
  3. 3Divide by 6: 36=s2.
  4. 4Take the square root: s=√36=6.
Answer: 6 cm

Why: Once the total surface area is divided among the six equal faces, the square root gives the side length of each face.

Section 8

Coordinate Geometry & Similar Figures

Geometry can involve points on a coordinate plane and shapes that have the same form but different sizes. A point is written (x,y): x gives horizontal position and y gives vertical position.

Horizontal and Vertical Distance

For points with the same y-coordinate, subtract their x-coordinates. For points with the same x-coordinate, subtract their y-coordinates.

Midpoint

The midpoint is halfway between two points. Use M=(x1+x22,y1+y22).

Average the x-coordinates and average the y-coordinates.

Scale Factor

Similar figures have the same shape but may have different sizes. Corresponding side lengths have the same ratio.

If every side doubles, the scale factor is 2 and the perimeter also doubles.

Similar Triangles

Similar triangles have proportional corresponding sides. If a side changes from 3 to 6, the scale factor is 63=2, so every corresponding length doubles.

Rules and formulas

  • Horizontal distance: subtract x-values.
  • Vertical distance: subtract y-values.
  • Midpoint: average corresponding coordinates.
  • Scale factor: new sideoriginal side.

Example 1

Find the midpoint between (2,4) and (8,10).

Solution

  1. 1Average the x-coordinates: 2+82=102=5.
  2. 2Average the y-coordinates: 4+102=142=7.
Answer: (5,7)

Why: The midpoint lies exactly halfway between the two points, so each coordinate is the average of the corresponding coordinates.

Example 2

Find the distance between (3,6) and (11,6).

Solution

  1. 1The y-coordinates are the same, so this is horizontal distance.
  2. 2Subtract the x-coordinates: 11-3=8.
Answer: 8 units

Why: When two points lie on the same horizontal line, only their x-values change.

Example 3

Similar triangles have corresponding sides 5 cm and 15 cm. Another side of the smaller triangle is 8 cm. Find the corresponding larger side.

Solution

  1. 1Find the scale factor: 155=3.
  2. 2The larger triangle is three times the size of the smaller triangle.
  3. 3Multiply the other side by 3: 8×3=24.
Answer: 24 cm

Why: Similar figures have proportional corresponding sides, so the same scale factor applies to every matching side.

Quick Review

Perimeter

  • Perimeter is distance around a shape.
  • Rectangle: P=2l+2w.
  • Square: P=4s.
  • Triangle: add all sides.

Area

  • Rectangle: A=lw.
  • Square: A=s2.
  • Triangle: A=12bh.
  • Circle: A=π r2.
  • Area uses square units.

Angles

  • Complementary: 90°.
  • Supplementary: 180°.
  • Triangle total: 180°.
  • Quadrilateral total: 360°.
  • Vertical angles are equal.

Pythagorean Theorem

  • For right triangles: a2+b2=c2.
  • c is the hypotenuse.
  • Common triples: 3-4-5, 5-12-13, and 8-15-17.

Circles

  • d=2r.
  • r=d2.
  • C=2π r.
  • C=π d.
  • A=π r2.

Volume

  • Rectangular prism: V=lwh.
  • Cube: V=s3.
  • Cylinder: V=π r2h.
  • Volume uses cubic units.

Surface Area

  • Cube: SA=6s2.
  • Rectangular prism: SA=2lw+2lh+2wh.
  • Surface area uses square units.

Coordinate Geometry & Similar Figures

  • Midpoint: average corresponding coordinates.
  • Horizontal distance: subtract x-values.
  • Vertical distance: subtract y-values.
  • Similar figures use the same scale factor for corresponding sides.