SAT Geometry & Trigonometry: method and traps

This is the smallest of the four Math areas and the most visual. A question that looks impossible in words is often obvious once drawn, and a sketch takes ten seconds. The formulas you need for area, volume and circles are supplied on the test; what is not supplied is the decision about which one applies.

Practise the 112 questions Play a timed round

This area at a glance

6 worked examples · 6 traps

112

questions

  • Foundation 21
  • Standard 64
  • Challenge 27

What this area asks for

  • Angle relationships, including parallel lines and transversals
  • Triangle angle sums and similar triangles
  • Area and perimeter of common figures
  • Circles: circumference, area, radius and diameter
  • Volume of prisms, cylinders, cones and spheres
  • Right triangles and the basic trigonometric ratios

The method

One routine that applies across the whole area, rather than a rule for each question type.

  1. 1

    Draw it, then label everything you know

    Even a rough sketch converts a sentence into a picture you can reason about. Write every given length and angle onto the drawing before doing arithmetic, and mark the quantity being asked for with a question mark.

  2. 2

    Halve or double before you substitute

    Circle questions give a diameter and want a radius, or the reverse, more often than they give what the formula needs. Convert first, then substitute, and the formula does the rest.

  3. 3

    Set up similar triangles as a single proportion

    When two figures are similar, corresponding sides share one scale factor. Find that factor once from the pair you know, then apply it to the side you want, rather than solving a proportion for each side.

  4. 4

    Remember which side a ratio uses

    Sine and cosine use the hypotenuse; tangent does not. Naming the three sides on your sketch before choosing a ratio removes the most common error in this area.

Worked examples

Each one is worked to the point where the next move is obvious, not to the answer alone.

1

A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?

  1. 1.Apply the Pythagorean theorem: 9^2 + 12^2 = 81 + 144 = 225.
  2. 2.Take the square root of 225.

15

2

A circle has circumference 18pi. What is its area in terms of pi?

  1. 1.Circumference is 2pi r, so 2pi r = 18pi and r = 9.
  2. 2.Area is pi r^2, so substitute r = 9.

81pi

3

A rectangle has its length and width each multiplied by 3. How does its area change?

  1. 1.Area is length times width, so the new area is (3l)(3w).
  2. 2.That equals 9 times lw.
  3. 3.Scaling every length by a factor multiplies area by the square of that factor.

The area becomes 9 times as large.

4

A circle has equation (x - 3)^2 + (y + 2)^2 = 49. What are its centre and radius?

  1. 1.The standard form is (x - h)^2 + (y - k)^2 = r^2, with centre (h, k).
  2. 2.Read the signs carefully: (y + 2) is (y - (-2)), so k is -2.
  3. 3.The right-hand side is r^2, not r, so take the square root: r = 7.

Centre (3, -2), radius 7

5

In a right triangle, one acute angle is 30 degrees and the hypotenuse is 20. How long is the side opposite the 30-degree angle?

  1. 1.Sine is opposite over hypotenuse, so opposite = hypotenuse x sin(30).
  2. 2.sin(30) is 1/2, a value worth knowing without a calculator.
  3. 3.20 x 1/2 = 10. This is also the 30-60-90 rule: the short leg is half the hypotenuse.

10

6

A cone and a cylinder have the same radius and the same height. What fraction of the cylinder's volume is the cone's?

  1. 1.Cylinder volume is pi r^2 h; cone volume is one third of pi r^2 h.
  2. 2.The radius and height are identical, so those factors cancel in the comparison.
  3. 3.What is left is the one third.

One third

Where marks are lost

Each of these has a matching wrong answer waiting among the four choices.

Using the diameter where the formula wants the radius

Substituting a diameter of 10 into pi r^2 gives 100pi instead of 25pi, and both appear as choices.

Scaling area and volume linearly

Doubling every length multiplies area by 4 and volume by 8, not by 2. This is the most reliable trap in the whole area.

Assuming a figure is drawn to scale

An angle that looks like a right angle is only a right angle if the question says so. Work from the labels, not the picture.

Picking the wrong trigonometric ratio

Tangent uses the two legs; sine and cosine each use the hypotenuse. A choice built on the wrong ratio is always among the four.

Scaling an area by the length factor

Doubling every side of a figure quadruples its area and multiplies its volume by eight. If a question gives a ratio of sides and asks about area, square it; if it asks about volume, cube it.

Adding a border once

A path of width w around a rectangle adds w at both ends of each dimension, so each dimension grows by 2w. Adding w once produces a smaller rectangle that still looks plausible, and the error only shows up in the final area.

Because the formulas are given, this area rewards setup over memory. If you can turn the sentence into a labelled drawing, the remaining work is arithmetic you already know how to do.

Now work the questions

112 Geometry and Trigonometry questions with a worked explanation for each, on the SAT Math page.

Open SAT Geometry & Trigonometry practice

The other SAT Math areas

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SAT® Geometry: Method and Practice | IQ Test Center