What the question looks like
Geometric comparison questions are a block within the Quantitative Skills section. Each one shows three small figures labeled (A), (B), and (C). They might be shapes divided into parts with some parts shaded, shapes with side lengths marked, or triangles with angles marked. Below the figures are four statements, and the instruction is simply to pick the true one.
The statements read like this:
- (A) is more shaded than (B).
- (A) and (C) are equally shaded.
- (B) is larger than (A).
- (C) is less shaded than (B).
On this page the figures are described in words instead of drawn. On the real paper test they are drawings, so the skill is turning a picture into a number, then comparing numbers.
Not every figure is a shape. The test also draws everyday objects and asks you to compare them: groups of coins by total value, measuring cups by how much liquid they hold, hourglasses by how long they have run, cards of dots by count or by the fraction filled, and bars on a graph by height or by a sum of bars. The method is the same. Turn each picture into one number, then compare the numbers, never the pictures.
How to solve it
- Decide what is being compared. Read the four choices first. They will all be about the same thing: shaded fraction, perimeter, area, or angle size.
- Turn each figure into one number. For shading, count the shaded parts and the total parts and write the fraction. For perimeter, add the sides. For area, multiply. Write the number next to the label.
- Put the three numbers in the same form. Reduce fractions or convert them to a common denominator so 2/8 and 1/4 look the same to you.
- Line the numbers up from smallest to largest, then read the choices again. Only one will match your order. Cross out choices as soon as they conflict with your numbers.
Do the counting on paper, not in your head. A figure with 8 parts and 2 shaded is easy to misread when you are moving fast.
Two angle facts settle most triangle comparisons without a protractor. The angles of a triangle add to 180 degrees, so if two are given, write the third on the figure before you read the choices; in a right triangle the other two add to 90. And the largest angle sits opposite the longest side, so in a triangle with sides 6, 6, and 9 the angle across from the 9 is the largest and the other two are equal.
Worked examples
Example 1
(A) a square divided into 4 equal parts with 1 part shaded. (B) a circle divided into 8 equal parts with 2 parts shaded. (C) a rectangle divided into 3 equal parts with 2 parts shaded. Which statement is true?
- A.(A) is more shaded than (B).
- B.(A) and (B) are equally shaded.
- C.(C) is less shaded than (A).
- D.(B) is more shaded than (C).
Show answer
Answer: B: (A) and (B) are equally shaded.
Write each shaded fraction: (A) is 1/4, (B) is 2/8, and (C) is 2/3. Since 2/8 reduces to 1/4, (A) and (B) are equally shaded, and (C) at 2/3 is the most shaded of the three. Choice A tempts students who count 2 shaded parts in (B) and assume more shaded pieces means more shading, but 2 of 8 is the same fraction as 1 of 4.
Example 2
(A) a square with each side 5 units. (B) a rectangle 8 units long and 2 units wide. (C) a triangle with sides 6, 7, and 7 units. Which statement is true?
- A.(A) has a greater perimeter than (B).
- B.(C) has the smallest perimeter.
- C.(A), (B), and (C) all have the same perimeter.
- D.(B) has a greater perimeter than (C).
Show answer
Answer: C: (A), (B), and (C) all have the same perimeter.
Perimeter is the distance around each shape. (A) is 5 + 5 + 5 + 5 = 20, (B) is 8 + 2 + 8 + 2 = 20, and (C) is 6 + 7 + 7 = 20. All three are equal, so choice C is true. Choice A is the trap: the square has the larger area, 25 compared with 16, but the question asks about perimeter, not area.
Example 3
(A) a triangle with angles of 60, 60, and 60 degrees. (B) a triangle with angles of 90, 45, and 45 degrees. (C) a triangle with angles of 100, 40, and 40 degrees. Which statement is true?
- A.The largest angle in (A) is greater than the largest angle in (B).
- B.(A) and (B) have equal largest angles.
- C.The largest angle in (B) is greater than the largest angle in (C).
- D.(C) has the largest angle of the three.
Show answer
Answer: D: (C) has the largest angle of the three.
Pick out the biggest angle in each figure: 60 in (A), 90 in (B), and 100 in (C). Ordering them gives (A) less than (B) less than (C), so (C) has the largest angle. Choice C tempts students who see the right angle in (B) and assume a right angle is the biggest kind of angle, but any angle above 90 degrees is larger.
Example 4
(A) a rectangle 6 units long and 4 units wide. (B) a square with each side 5 units. (C) a triangle with a base of 8 units and a height of 6 units. Which statement is true?
- A.(A) and (C) have equal areas, and (B) is larger than both.
- B.(B) is smaller than (A).
- C.(A), (B), and (C) all have the same area.
- D.(C) is larger than (B).
Show answer
Answer: A: (A) and (C) have equal areas, and (B) is larger than both.
Compute each area. (A) is 6 × 4 = 24, (B) is 5 × 5 = 25, and (C) is 1/2 × 8 × 6 = 24. So (A) and (C) are equal and (B) is larger by one square unit. Choice D is the trap for students who forget the 1/2 in the triangle formula and get 48.
Traps to watch for
- Counting pieces instead of fractions. Two shaded parts out of 8 is less shading than one shaded part out of 3. Always compare the fraction, never the raw count.
- Mixing up perimeter and area. A long thin rectangle can have a big perimeter and a small area. Read the choices to see which one the question is about before you start calculating.
- Trusting your eyes. A circle and a square divided into the same number of parts can look different in size on the page. Use the numbers, not the drawing.
- Stopping after the first true-looking choice. Check that your numbers rule out the other three. If two choices both seem true, you have a counting error somewhere.
Pacing
These questions take a little longer than number series because there are three figures to read, but aim to finish each one in about 35 seconds. If you have written down the three numbers and still cannot make a choice fit, pick the option that matches most of your work and keep moving. The HSPT has no guessing penalty, so never leave one blank.
Practice
See how you do on geometric comparison under a real clock with our free diagnostic.
Shaky on the skill itself? Start with the lessons on geometry, measurement and units, and data and probability, then come back to the question format.