Area of Common Figures — Calculate area of triangles, parallelograms, and trapezoids. - SHSAT Math

SECTION 1

Why Do We Measure Area?

Thousands of years ago, people needed to measure land. Farmers had to know how much ground they owned. Tax collectors needed to figure out what people owed. The answer to all of these questions came down to one idea: area (the amount of flat space inside a shape).

Ancient civilizations discovered formulas for area step by step. They started with rectangles, then figured out triangles, parallelograms, and trapezoids. These same formulas still show up on tests like the SHSAT today!

~3000 BCE • Ancient Egypt

Egyptian surveyors measured farmland along the Nile River after yearly floods washed away boundary markers. They used ropes and stakes to calculate areas of rectangular and triangular plots.

~2000 BCE • Babylonian Math

Babylonian scribes wrote area formulas on clay tablets. They could find the area of trapezoids, which helped them divide land fairly among workers.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid organized all known geometry into a famous book. He proved why each area formula works using logical reasoning.

Today • Modern Testing

Area formulas appear on the SHSAT and other standardized tests. Architects, engineers, and designers use these same formulas every day.

The big question these formulas answer is simple: How much space does a flat shape cover? Let's learn the three formulas you need.

SECTION 2

Core Principles of Area

Before jumping into formulas, you need to understand a few key ideas. These principles connect all three shapes we will study.

Area Is Measured in Square Units

Area counts how many unit squares fit inside a shape. That is why we write square units like cm², in², or ft².

Base and Height Must Be Perpendicular

The base is one side of the shape. The height is the straight-up distance from the base to the top. They form a 90° angle.

Every Shape Connects to a Rectangle

A triangle is half a rectangle. A parallelogram can be rearranged into a rectangle. A trapezoid is the average of two rectangles. This is the secret behind all three formulas.

Height Is Not Always a Side

Be careful! The height is the perpendicular (straight-up) distance, not a slanted side. On the SHSAT, this detail is a common trick.

✦

KEY TAKEAWAY

Think of area like painting a wall. The base is how wide your roller goes, and the height is how tall you paint. The area tells you how much paint you need. Just remember: the roller always moves straight up, never on a slant!

SECTION 3

Seeing the Three Shapes

The diagram below shows all three shapes side by side. Notice how each one labels the base (b) and the height (h). The dashed line is the height — it always makes a right angle with the base.

All three shapes use a base and a height. The dashed line shows the height, which is always perpendicular (at 90°) to the base. The trapezoid has two bases: b₁ (bottom) and b₂ (top).

Look at the dashed lines carefully. In each shape, the height drops straight down at a 90° angle. The little square in the corner proves it. On the SHSAT, you might see shapes that are tilted or slanted. Don't let that trick you — always look for the perpendicular height.

SECTION 4

The Three Area Formulas

Here are the three formulas you need to memorize. Each one is simple, but understanding why it works helps you remember it.

AREA OF A TRIANGLE

A = ½ × b × h

A = area, b = length of the base, h = perpendicular height. A triangle is exactly half of a rectangle with the same base and height, so we multiply by ½.

AREA OF A PARALLELOGRAM

A = b × h

A = area, b = length of the base, h = perpendicular height. If you cut off a triangle from one side and slide it to the other, you get a rectangle. That is why the formula is the same as a rectangle's.

AREA OF A TRAPEZOID

A = ½ × (b₁ + b₂) × h

A = area, b₁ = length of the first base (usually bottom), b₂ = length of the second base (usually top), h = perpendicular height. You add the two bases, take their average (divide by 2), then multiply by the height.

💡

SHSAT Tip

The SHSAT usually gives you the formulas on a reference sheet. But knowing them by heart saves you time and helps you spot which formula to use faster.

SECTION 5

Why Each Formula Works

The best way to remember a formula is to understand the picture behind it. The diagram below shows how each shape connects to a simple rectangle.

Top-left: A triangle fills exactly half a rectangle. Top-right: Slide the cut piece of a parallelogram to form a rectangle. Bottom: A trapezoid's area equals its average width times the height.

In the top-left picture, the dashed rectangle has area b × h. The shaded triangle fills exactly half, so its area is ½ × b × h. In the top-right picture, the gold triangle on the left side of the parallelogram slides to the right. Now it is a rectangle with area b × h. The bottom picture shows that a trapezoid acts like a shape whose width is the average of its two bases. So the area is ½ × (b₁ + b₂) × h.

SECTION 6

Worked Examples

Example 1 — Triangle

1
Step 1 — Read the problem

A triangle has a base of 10 cm and a height of 6 cm. Find its area.
2
Step 2 — Write the formula

A = ½ × b × h
3
Step 3 — Substitute the values

A = ½ × 10 × 6
4
Step 4 — Calculate

First, 10 × 6 = 60. Then, ½ × 60 = 30.

A = 30 cm²

Example 2 — Parallelogram

1
Step 1 — Read the problem

A parallelogram has a base of 12 in and a height of 5 in. A slanted side measures 7 in. Find its area.
2
Step 2 — Identify base and height (ignore the slant!)

b = 12 in, h = 5 in. The slanted side (7 in) is not used in the formula.
3
Step 3 — Substitute and calculate

A = b × h = 12 × 5 = 60

A = 60 in²

Example 3 — Trapezoid

1
Step 1 — Read the problem

A trapezoid has bases of 8 ft and 14 ft and a height of 5 ft. Find its area.
2
Step 2 — Write the formula

A = ½ × (b₁ + b₂) × h
3
Step 3 — Substitute the values

A = ½ × (8 + 14) × 5
4
Step 4 — Simplify inside parentheses first

8 + 14 = 22, so A = ½ × 22 × 5
5
Step 5 — Calculate

½ × 22 = 11. Then 11 × 5 = 55.

A = 55 ft²

SECTION 7

Comparing the Three Formulas

All three formulas use a base and height, but each shape has its own twist. The table below helps you see the differences at a glance.

Shape	Formula	Key Detail
Triangle	A = ½ × b × h	Don't forget the ½. Height is perpendicular to the base, not a slanted side.
Parallelogram	A = b × h	Same as a rectangle. Use the perpendicular height, NOT the slanted side.
Trapezoid	A = ½ × (b₁ + b₂) × h	Has TWO bases (the parallel sides). Add them first, then divide by 2, then multiply by the height.

Quick-reference comparison of the three area formulas.

⚠️

COMMON MISTAKE ALERT

Imagine you are measuring a leaning bookshelf. The height is how tall it is from the floor straight up to the top — not the length of the slanted side. The same rule applies to parallelograms and trapezoids. If a problem gives you a slanted side, it is probably trying to trick you!

SECTION 8

Connecting to Harder Topics

Once you master these three shapes, you will be ready for more advanced problems. On the SHSAT and in high school geometry, you will see composite figures (shapes made by combining simpler shapes). You might also work with shaded region problems where you subtract one area from another.

What You Know Now	What Comes Next
Area of a single triangle	Break an irregular polygon into multiple triangles to find its total area
Area of a parallelogram	Find the area of rhombuses and other special parallelograms using diagonals
Area of a trapezoid	Use the trapezoid formula in coordinate geometry problems (with graphed vertices)
All three formulas	Surface area of 3-D solids (prisms, pyramids) that use triangle and trapezoid faces

How today's formulas connect to future topics.

The key idea is that almost every area problem, no matter how complex, can be broken down into triangles, rectangles, parallelograms, and trapezoids. Mastering these building blocks makes harder problems much easier.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A parallelogram and a rectangle have the same base and the same height. Which shape has a larger area, or are they equal? Explain your reasoning.

Show answer

They have the same area. Both use the formula A = b × h. A parallelogram can be rearranged into a rectangle by cutting a triangle off one end and sliding it to the other. If the base and height are equal, the area must be equal too.

Question 2: BASIC CALCULATION

Find the area of a triangle with a base of 16 cm and a height of 9 cm.

Show answer

Use the formula A = ½ × b × h. Substitute: A = ½ × 16 × 9. First, 16 × 9 = 144. Then ½ × 144 = 72. The area is 72 cm².

Question 3: INTERMEDIATE

A trapezoid has an area of 60 in². Its two bases measure 8 in and 12 in. What is the height of the trapezoid?

Show answer

Start with A = ½ × (b₁ + b₂) × h. Substitute: 60 = ½ × (8 + 12) × h. Simplify inside the parentheses: 8 + 12 = 20. So 60 = ½ × 20 × h, which means 60 = 10 × h. Divide both sides by 10: h = 6. The height is 6 in.

Question 4: APPLIED

A park has the shape of a parallelogram. Its base is 120 meters and its height is 45 meters. The park managers want to plant grass that costs $3 per square meter. How much will it cost to cover the entire park?

Show answer

First, find the area: A = b × h = 120 × 45 = 5,400 m². Next, multiply the area by the cost per square meter: 5,400 × $3 = $16,200. It will cost $16,200 to cover the park. This is a two-step problem: find the area first, then use it to find the total cost.

Question 5: CRITICAL THINKING

A large triangle has a base of 20 ft and a height of 12 ft. A smaller trapezoid is cut out from inside the triangle. The trapezoid has bases of 6 ft and 10 ft and a height of 4 ft. What is the area of the remaining (shaded) region?

Show answer

Find the area of the triangle: A = ½ × 20 × 12 = 120 ft². Find the area of the trapezoid: A = ½ × (6 + 10) × 4 = ½ × 16 × 4 = 32 ft². Subtract the trapezoid from the triangle: 120 − 32 = 88. The shaded region is 88 ft². This is a composite figure problem — you find each area separately, then subtract. These show up often on the SHSAT.

SUMMARY

Lesson Summary

Summary

You learned three essential area formulas. The area of a triangle is ½ × b × h because a triangle is half a rectangle. The area of a parallelogram is b × h because it rearranges into a rectangle. The area of a trapezoid is ½ × (b₁ + b₂) × h because you average its two parallel bases.

The most important rule across all three formulas: the height is always the perpendicular distance from the base to the opposite side — never a slanted edge. Watch for this on the SHSAT. With these three formulas, you can also tackle composite figure problems by breaking complex shapes into simpler pieces.