Compare Geometric Figures | Varsity Tutors

SECTION 1

Historical Context & Motivation

People have been comparing shapes for thousands of years. Ancient builders needed to know which plot of land was bigger, or how much stone to cut for a wall. Comparing geometric figures means looking at two or more shapes and deciding which one is larger, smaller, or equal based on measurements like area, perimeter, or volume.

This skill shows up on the HSPT Quantitative section. You will see questions that ask you to compare two columns — Column A and Column B — where each column contains a geometric measurement. Your job is to figure out which is greater, or if they are equal.

~2000 BCE

Egyptian Land Surveying

Ancient Egyptians compared field areas after the Nile flooded each year. They used rope stretchers to measure and compare plots of land.

~300 BCE

Euclid's Elements

The Greek mathematician Euclid wrote rules for comparing shapes. His book showed how to prove that two triangles are equal or that one circle is bigger than another.

~250 BCE

Archimedes and Area

Archimedes found ways to compare curved shapes to straight ones. He proved the area of a circle by comparing it to a triangle.

Today

Standardized Testing

Modern tests like the HSPT ask students to compare geometric figures quickly and accurately using formulas for area, perimeter, and volume.

The big question this lesson answers is: How do you decide which of two geometric figures has a greater area, perimeter, or other measurement? Let's find out!

SECTION 2

Core Principles & Definitions

Before you can compare figures, you need to know the key measurements. Think of these as the "stats" of each shape — just like a basketball player has points, rebounds, and assists, a shape has its own numbers that describe it.

Area

Area is the amount of flat space inside a shape. It is measured in square units, like in² or cm². A bigger area means the shape covers more space.

Perimeter / Circumference

Perimeter is the total distance around the outside of a shape. For circles, we call this the circumference. It is measured in regular units like inches or centimeters.

Volume

Volume is the amount of space inside a 3D (three-dimensional) shape. It is measured in cubic units, like in³ or cm³. Think of it as how much water a shape could hold.

Comparing Columns

On the HSPT, you compare Column A to Column B. Your answer choices are: A is greater, B is greater, they are equal, or you cannot determine.

✦ KEY TAKEAWAY

Think of comparing geometric figures like comparing two pizzas. One pizza might have a bigger surface (area) but the other might have a longer crust edge (perimeter). You have to know which measurement the question is asking about before you can decide which figure "wins."

SECTION 3

Visual Explanation

The diagram below shows four common shapes side by side. Each shape has its area formula written inside and its perimeter formula written below. Notice how different shapes with the same side length can have very different areas.

Four common shapes with their area and perimeter formulas. The bottom panel highlights that the circle has the largest area, showing why you must always calculate before comparing.

As you can see, a circle with a radius of just 5 has a much bigger area than a square with side 6. This is why you should never guess which figure is larger just by looking at it. Always use the formulas to calculate the exact value before comparing.

SECTION 4

Key Formulas You Need

On the HSPT, you will use a handful of formulas over and over. Let's go through each one so you know exactly when to use it.

AREA OF A RECTANGLE

A = l × w

Where l = length and w = width. For a square, l and w are the same, so A = s².

AREA OF A TRIANGLE

A = ½ × b × h

Where b = base and h = height. The height must be perpendicular (straight up) from the base.

AREA OF A CIRCLE

A = π × r²

Where r = radius (the distance from the center to the edge). Use π ≈ 3.14 on the HSPT.

CIRCUMFERENCE OF A CIRCLE

C = 2 × π × r

This is the distance around the circle. You can also write it as C = π × d, where d = diameter (twice the radius).

💡 HSPT Tip

You usually do NOT need an exact decimal answer on comparison questions. Often you can tell which value is bigger after just one or two steps of calculation. Save time by estimating when possible!

SECTION 5

Strategies for Comparing Figures

Now that you know the formulas, let's talk about how to actually compare two figures step by step. The diagram below shows a typical HSPT-style comparison between Column A and Column B.

A step-by-step walkthrough of an HSPT-style comparison. First, identify what is being compared. Then choose the right formula, plug in numbers, and compare the results.

Read carefully. Are you comparing area, perimeter, or something else? Don't mix them up!
Write the formula. Pick the correct formula for each shape. This prevents silly mistakes.
Substitute and simplify. Plug in the numbers, then solve. Use 3.14 for π unless told otherwise.
Compare the two results. Use >, <, or = to state your answer.

SECTION 6

Worked Example

Let's work through a full example together. This is the kind of question you will see on the HSPT.

📝 SAMPLE QUESTION

Column A: The perimeter of a rectangle with length 12 and width 5. Column B: The circumference of a circle with diameter 10. Which is greater?

Comparing Perimeter vs. Circumference

Step 1 — Identify the MeasurementBoth columns ask about the distance around the outside of a shape. For the rectangle, that is perimeter. For the circle, that is circumference.

Step 2 — Write the FormulasRectangle perimeter: P = 2(l + w). Circle circumference: C = π × d.

Step 3 — Plug in the Numbers for Column AP = 2(12 + 5) = 2(17) = 34

Column A = 34

Step 4 — Plug in the Numbers for Column BC = 3.14 × 10 = 31.4

Column B ≈ 31.4

Step 5 — Compare34 > 31.4, so Column A is greater.

Answer: Column A is greater.

SECTION 7

Common Traps & Tips

The HSPT is designed to test how carefully you read. Here are the most common mistakes students make when comparing geometric figures, and how to avoid them.

Common mistakes on HSPT geometric comparison questions

Common Trap	Why It Happens	How to Avoid It
Mixing up area and perimeter	Both involve the same numbers, so it's easy to grab the wrong formula.	Underline the word "area" or "perimeter" in the question before you start.
Confusing radius and diameter	The question says "diameter 10" but you use r = 10 by mistake.	Always check: radius = half the diameter. If d = 10, then r = 5.
Forgetting to square the radius	You write πr instead of πr². This gives a much smaller area.	Write the formula first, then substitute. The exponent reminds you.
Assuming bigger sides = bigger area	A long, thin rectangle can have less area than a short, wide one.	Always calculate both values. Never assume based on one dimension.

✦ KEY TAKEAWAY

Think of it like comparing two athletes. One might be taller (bigger perimeter), but the other might weigh more (bigger area). You can't compare height to weight — they measure different things. Always make sure you're comparing the same type of measurement for both figures.

SECTION 8

Connecting to Advanced Geometry

On the HSPT, most comparison questions involve 2D (flat) shapes. But in high school geometry, you will also compare 3D shapes using volume and surface area. The same strategy works: identify what you are comparing, use the right formula, plug in numbers, and compare.

HSPT vs. high school geometry comparisons

HSPT Level (This Lesson)	High School Geometry (Coming Soon)
Compare areas of flat shapes (rectangles, triangles, circles)	Compare volumes of 3D shapes (prisms, cylinders, cones, spheres)
Compare perimeters and circumferences	Compare surface areas of 3D shapes
Use simple numbers and π ≈ 3.14	Leave answers in terms of π; use variables
Compare two specific figures at a time	Use similarity ratios to compare families of shapes

If you master comparing 2D figures now, you will have a huge head start when you encounter 3D comparisons in high school. The thinking process is exactly the same — only the formulas get a little longer.

SECTION 9

Practice Problems

Try these five problems on your own. They get harder as you go. For each, decide whether Column A is greater, Column B is greater, or they are equal.

PROBLEM 1 — CONCEPTUAL

Column A: The area of a square with side 5. Column B: The area of a rectangle with length 5 and width 5. Which is greater, or are they equal?

PROBLEM 2 — BASIC CALCULATION

Column A: The perimeter of a square with side 9. Column B: The perimeter of an equilateral triangle with side 12. Which is greater?

PROBLEM 3 — INTERMEDIATE

Column A: The area of a triangle with base 14 and height 10. Column B: The area of a circle with radius 7. Use π ≈ 3.14. Which is greater?

PROBLEM 4 — APPLIED

A school is painting two walls. Wall A is a rectangle that is 20 feet long and 8 feet tall. Wall B is a square that is 13 feet on each side. Column A: The area of Wall A. Column B: The area of Wall B. Which wall needs more paint?

PROBLEM 5 — CRITICAL THINKING

Column A: The area of a circle with radius 6. Column B: The area of a square whose perimeter equals the circumference of that same circle. Use π ≈ 3.14. Which is greater?

SUMMARY

Lesson Summary

Comparing geometric figures on the HSPT means calculating a specific measurement — like area, perimeter, or circumference — for each figure and then deciding which is greater, or if they are equal. The key formulas you need are A = l × w for rectangles, A = ½ × b × h for triangles, and A = π × r² for circles.

Always follow the four-step process: identify what you are comparing, choose the right formula, plug in the numbers, and compare the results. Watch out for common traps like mixing up radius and diameter or confusing area with perimeter. Calculate first, then compare — never guess!