Calculate area of rectangles and triangles. - SSAT Middle Level Quantitative

SECTION 1

Where Did Area Formulas Come From?

People have measured area (the amount of flat space inside a shape) for thousands of years. Ancient farmers needed to know the size of their fields so they could divide land fairly and pay the right amount of tax. Builders needed area calculations to plan temples, pyramids, and homes.

Over time, mathematicians in Egypt, Babylon, Greece, and beyond figured out simple rules — or formulas — that let anyone find area quickly. These same formulas are the ones you will use today!

~3000 BCE • Ancient Egypt

Egyptian surveyors measured rectangular farm plots along the Nile River after yearly floods washed away boundary markers.

~1800 BCE • Babylonian Clay Tablets

Babylonian scribes recorded formulas for areas of rectangles and triangles on clay tablets, some of which survive today.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid organized geometry into a logical system. He proved why the area formulas work, not just how to use them.

Today • Everyday Use

We use area every day — buying carpet, painting walls, designing phone screens, and solving SSAT problems!

The big question these early thinkers answered was: How can we figure out the size of a flat shape without covering it in tiny squares one by one? The answer? Clever formulas that use just a couple of measurements.

SECTION 2

Core Principles & Definitions

Before jumping into formulas, let's nail down a few key ideas. Understanding these will make every area problem feel easier.

Area = Inside Space

Area is the number of square units that fit inside a flat (two-dimensional) shape. We measure it in square units like in², cm², or ft².

Base & Height

The base (b) is one side of the shape. The height (h) is the perpendicular (straight-up) distance from the base to the opposite side or vertex.

Perpendicular Means 90°

Perpendicular means the height meets the base at a right angle (90°). This is very important for triangles — the height is NOT always a side of the triangle.

Square Units

Area is always written in square units. If sides are in inches, area is in square inches (in²). If sides are in centimeters, area is in cm².

✦

KEY TAKEAWAY

Think of area like tiling a floor. If your bathroom is 5 feet long and 3 feet wide, you could lay out rows of 1-foot square tiles: 5 tiles across and 3 rows deep, giving you 5 × 3 = 15 tiles total. That's 15 square feet of area. A triangle is exactly half of a rectangle, so its formula uses the same idea but divides by 2.

SECTION 3

Seeing Area: Rectangles & Triangles

The diagram below shows a rectangle made of square units. Count the squares inside — that's the area! You can also just multiply the number of columns (base) by the number of rows (height).

A rectangle with base 6 and height 4. Each small square is one square unit. The shaded top row shows 6 squares; there are 4 rows total, so the area is 6 × 4 = 24.

Now look at the next diagram. It shows how a triangle is exactly half of a rectangle. If you draw a diagonal line across a rectangle, you split it into two equal triangles. That's why the triangle formula divides by 2.

Left: A rectangle split along its diagonal into two equal triangles. Right: A standalone triangle with base 10 and height 8. Its area is half of 10 × 8 = 40 square units.

SECTION 4

The Formulas You Need

There are only two formulas to remember for this topic. Let's look at each one and understand what every part means.

AREA OF A RECTANGLE

A = b × h

A = area (in square units), b = base (length of the bottom side), h = height (length of a side perpendicular to the base). For a rectangle, you can also call these length and width.

AREA OF A TRIANGLE

A = ½ × b × h

A = area (in square units), b = base (any side of the triangle), h = height (the perpendicular distance from the base to the opposite vertex). Remember: the height must form a 90° angle with the base.

💡

Why divide by 2?

A triangle is exactly half of a rectangle that has the same base and height. That's where the ½ comes from. If you ever forget the triangle formula, just think: find the rectangle's area, then cut it in half.

On the SSAT, the numbers are usually whole numbers or simple decimals. You won't need a calculator — the math is designed to work out neatly.

SECTION 5

Finding the Right Height

The most common mistake students make is using a slanted side of a triangle instead of the true perpendicular height. Let's look at three types of triangles and where the height lives in each one.

Three triangles showing where the height is located. In a right triangle, the height is one of the legs. In acute and obtuse triangles, the height drops from the top vertex straight down to the base.

Triangle Type	Where Is the Height?	SSAT Tip
Right triangle	One of the two legs (sides that form the right angle) IS the height.	Use the two legs as base and height. The longest side (hypotenuse) is NOT the height.
Acute triangle	The height drops from the top vertex down inside the triangle to the base.	Look for a dashed line with a right-angle square at the base.
Obtuse triangle	The height may fall inside the triangle or outside if the base is extended.	The problem will always tell you or show you the height. Don't guess — use the given number.

Height locations for different triangle types

SECTION 6

Step-by-Step Worked Examples

Example 1: Rectangle Area

1
Step 1 — Read the problem

A rectangular garden is 12 feet long and 7 feet wide. What is its area?
2
Step 2 — Identify the base and height

Base (b) = 12 ft, Height (h) = 7 ft. For a rectangle, the length and width play the roles of base and height.
3
Step 3 — Substitute into the formula

A = b × h = 12 × 7
4
Step 4 — Calculate

12 × 7 = 84

A = 84 ft²

Example 2: Triangle Area

1
Step 1 — Read the problem

A triangle has a base of 10 cm and a height of 6 cm. What is its area?
2
Step 2 — Identify the base and height

Base (b) = 10 cm, Height (h) = 6 cm.
3
Step 3 — Substitute into the formula

A = ½ × b × h = ½ × 10 × 6
4
Step 4 — Multiply step by step

First, 10 × 6 = 60. Then divide by 2: 60 ÷ 2 = 30.

A = 30 cm²

💡

Quick Tip

When multiplying by ½, you can either multiply the two numbers first and divide by 2, or divide one of the numbers by 2 first and then multiply. For example, ½ × 10 × 6 can be solved as (10 ÷ 2) × 6 = 5 × 6 = 30. Pick whichever order makes the mental math easier!

SECTION 7

Common Mistakes & How to Avoid Them

Even though the formulas are short, there are a few traps that trip students up on the SSAT. Let's go through them so you don't fall in!

Mistake	Why It's Wrong	Fix
Forgetting to divide by 2 for a triangle	A triangle is half a rectangle, so b × h gives you double the correct answer.	Always check: is it a triangle? If yes, divide by 2.
Using a slanted side as the height	The height must be perpendicular (at a 90° angle) to the base. Slant sides are longer than the true height.	Look for the right-angle mark (little square) in the diagram.
Mixing up area and perimeter	Perimeter is the distance around a shape (add the sides). Area is the space inside (multiply).	Read the question carefully: does it say 'area' or 'perimeter'?
Forgetting square units	Area is measured in square units (ft², cm²), not plain units (ft, cm).	After you compute the number, write the unit with a little 2 (squared).

Watch out for these on the SSAT!

✦

KEY TAKEAWAY

Think of the height like a plumb line a builder drops from the roof straight down to the ground. It doesn't lean — it goes straight down at 90°. If the line would lean at all, it's not the height — it's a slant.

SECTION 8

Connecting to Bigger Ideas

The area formulas for rectangles and triangles are building blocks for almost every other area formula you'll meet later in math. Here's a peek at how they connect.

Shape	Formula	How It Relates
Rectangle	A = b × h	The foundation. Every shape can be broken into rectangles.
Triangle	A = ½ × b × h	Half of a rectangle. Many polygons split into triangles.
Parallelogram	A = b × h	Same as a rectangle! You can rearrange a parallelogram into a rectangle.
Trapezoid	A = ½ × (b₁ + b₂) × h	Uses the average of two bases — still built from triangles and rectangles.
Circle	A = π × r²	Can be thought of as many tiny triangles arranged around a center point.

Area formulas build on each other

On the SSAT, you mostly need rectangles and triangles. But sometimes a problem gives you an irregular shape and asks you to break it into rectangles and triangles, find each area, and add them up. Knowing these two formulas well prepares you for that.

ℹ

Looking Ahead

In later math classes, you'll learn to find the area under curved lines using a process called integration. The amazing thing? It still works by slicing the curve into tons of tiny rectangles and triangles and adding them up — the same basic idea you're learning now.

SECTION 9

Practice Problems

Try these five problems. They get harder as you go. For each one, pick the best answer from (A) through (E).

Question 1: CONCEPTUAL

Why does the formula for the area of a triangle include dividing by 2? (A) Because a triangle has 2 sides (B) Because a triangle is half of a rectangle with the same base and height (C) Because you always divide area by 2 (D) Because a triangle has 2 angles that are equal (E) Because the height is half the base

Show answer

The correct answer is (B). A triangle is exactly half of a rectangle that shares the same base and height. If you draw a diagonal across a rectangle, you get two identical triangles, so each triangle's area is half the rectangle's area. Choice (A) is wrong because having 2 sides doesn't relate to the formula. Choice (C) is wrong because you don't divide by 2 for every shape — only triangles. Choice (D) is wrong because a triangle's angles don't have to be equal. Choice (E) is wrong because the height and base are independent measurements.

Question 2: BASIC CALCULATION

A rectangle has a length of 9 inches and a width of 5 inches. What is its area? (A) 14 in² (B) 28 in² (C) 45 in² (D) 45 in (E) 22.5 in²

Show answer

The correct answer is (C). Using the formula A = b × h, we get A = 9 × 5 = 45 in². Choice (A) gives 14, which is 9 + 5 — that's the sum of two sides, not the area. Choice (B) gives 28, which is the perimeter 2 × (9 + 5). Choice (D) gives the right number (45) but uses the wrong unit — area must be in square inches (in²), not plain inches. Choice (E) gives 22.5, which is what you'd get if you divided by 2 — that's the triangle formula, not the rectangle formula.

Question 3: INTERMEDIATE

A triangle has a base of 14 cm and a height of 9 cm. What is the area of the triangle? (A) 23 cm² (B) 63 cm² (C) 126 cm² (D) 46 cm² (E) 31.5 cm²

Show answer

The correct answer is (B). Using the formula A = ½ × b × h, we substitute: A = ½ × 14 × 9. First, 14 × 9 = 126. Then, 126 ÷ 2 = 63. So the area is 63 cm². Choice (A) is 14 + 9 = 23, which adds the measurements instead of multiplying. Choice (C) is 126, which is the full rectangle area without dividing by 2. Choice (D) is 46, which doesn't come from the correct formula. Choice (E) is 31.5, which would result from dividing by 4 instead of 2.

Question 4: APPLIED

Maria wants to paint one wall of her room. The wall is a rectangle that is 12 feet wide and 8 feet tall, but it has a triangular window with a base of 4 feet and a height of 3 feet. What is the area she needs to paint? (A) 90 ft² (B) 96 ft² (C) 84 ft² (D) 102 ft² (E) 78 ft²

Show answer

The correct answer is (A). First, find the wall's area: A = 12 × 8 = 96 ft². Then find the window's area: A = ½ × 4 × 3 = 6 ft². Subtract the window from the wall: 96 − 6 = 90 ft². Choice (B) is 96, the wall area without subtracting the window. Choice (C) is 84, which subtracts 12 instead of 6 (forgetting to halve the triangle). Choice (D) is 102, which adds the window area instead of subtracting it. Choice (E) is 78, which subtracts too much (perhaps subtracting 4 × 3 = 12 and making an additional error).

Question 5: CRITICAL THINKING

A rectangle and a triangle have the same base of 10 inches. The rectangle has a height of 6 inches. If the triangle has the same area as the rectangle, what is the triangle's height? (A) 6 inches (B) 3 inches (C) 12 inches (D) 10 inches (E) 30 inches

Show answer

The correct answer is (C). The rectangle's area is 10 × 6 = 60 in². For the triangle to have the same area, we need ½ × 10 × h = 60. That simplifies to 5 × h = 60, so h = 60 ÷ 5 = 12 inches. Since a triangle only uses half of b × h, its height must be twice the rectangle's height to match the same area. Choice (A) gives 6, which assumes the heights are equal — but then the triangle's area would only be 30 in². Choice (B) gives 3, which is half of 6, making the triangle even smaller. Choice (D) gives 10, which confuses height with base. Choice (E) gives 30, which is the area itself, not the height.

SUMMARY

Lesson Summary

Summary

Area measures the flat space inside a shape, always expressed in square units. For a rectangle, multiply the base × height (A = b × h). For a triangle, use A = ½ × b × h because a triangle is exactly half of a rectangle with the same base and height.

Always make sure the height is perpendicular to the base — look for the right-angle mark in diagrams. Don't confuse area (space inside) with perimeter (distance around). On the SSAT, read carefully, identify which shape you have, pick the right formula, plug in the numbers, and remember your square units.