Compute perimeter and area of common plane figures - TACHS Math

SECTION 1

Why Do We Measure Shapes?

People have been measuring shapes for thousands of years. Ancient farmers needed to know how much land they owned. Builders needed to figure out how much material to use for walls and floors. These everyday problems led to the math we now call geometry (the study of shapes, sizes, and spaces).

Two of the most useful measurements are perimeter (the total distance around a shape) and area (the amount of flat space a shape covers). Let's look at how these ideas developed over time.

3000 BCE • Ancient Egypt

Egyptian farmers measured fields along the Nile River after yearly floods washed away boundary markers. They used ropes to find distances and calculate areas for taxes.

300 BCE • Euclid's Elements

The Greek mathematician Euclid wrote a famous textbook that organized geometry rules. His formulas for area and perimeter are still used today.

250 BCE • Archimedes and the Circle

Archimedes figured out how to calculate the area and circumference of a circle. He estimated the value of π (pi) to a remarkable level of accuracy.

Today • Modern Applications

We use perimeter and area every day — from buying carpet for a room to designing a phone screen. These formulas appear on tests like the TACHS.

The big question that geometry answers is: How do we describe the size of a flat shape using numbers? Perimeter and area give us two different ways to do exactly that.

SECTION 2

Core Principles & Definitions

Before we jump into formulas, you need to understand a few key ideas. These principles apply to every shape you will work with.

Perimeter = Distance Around

Imagine walking along the edges of a shape and back to where you started. The total distance you walk is the perimeter. It is measured in linear units like feet, meters, or inches.

Area = Space Inside

Area tells you how many unit squares fit inside a shape. It is measured in square units like ft², m², or in².

Units Matter

Perimeter uses regular units (cm, ft). Area uses squared units (cm², ft²). Always include the correct unit in your answer.

Know Your Dimensions

Most formulas need a length, width, height, base, or radius. Label and identify these measurements before plugging into a formula.

✦

KEY TAKEAWAY

Think of perimeter like the fence around a yard — it measures the border. Think of area like the grass inside the yard — it measures the surface. A yard can have a short fence but lots of grass, or a long fence but little grass. Perimeter and area measure two completely different things.

SECTION 3

Seeing Perimeter and Area

The diagram below shows a rectangle, a triangle, and a circle side by side. The colored border represents the perimeter (or circumference) of each shape, and the shaded interior represents the area.

The violet border shows the rectangle's perimeter, the cyan border shows the triangle's sides, and the pink border shows the circle's circumference. The shaded fill in each shape represents its area.

Notice how each shape uses a different formula. The rectangle multiplies length times width. The triangle uses half the base times the height. The circle uses π (pi), which is about 3.14. In the next section, we will list every formula you need.

SECTION 4

The Formulas You Need

Here are the key formulas for the shapes you will see on the TACHS. For each one, we show both the perimeter (or circumference) and the area formula.

Rectangle

PERIMETER OF A RECTANGLE

P = 2l + 2w

l = length (the longer side), w = width (the shorter side). Add both lengths and both widths.

AREA OF A RECTANGLE

A = l × w

Multiply length times width. The answer is in square units (like cm² or ft²).

Square

PERIMETER OF A SQUARE

P = 4s

s = the length of one side. All four sides are equal, so you just multiply by 4.

AREA OF A SQUARE

A = s²

Multiply the side by itself: s × s.

Triangle

PERIMETER OF A TRIANGLE

P = a + b + c

a, b, c are the three side lengths. Add them all up.

AREA OF A TRIANGLE

A = ½ × b × h

b = base, h = height (measured straight up from the base to the top). Multiply base times height, then divide by 2.

Circle

CIRCUMFERENCE (PERIMETER) OF A CIRCLE

C = 2πr or C = πd

r = radius (center to edge), d = diameter (edge to edge through center). π ≈ 3.14.

AREA OF A CIRCLE

A = πr²

Square the radius first, then multiply by π.

Parallelogram & Trapezoid

AREA OF A PARALLELOGRAM

A = b × h

b = base, h = height (the perpendicular distance between the top and bottom sides). Do NOT use the slanted side as the height.

AREA OF A TRAPEZOID

A = ½ × (b₁ + b₂) × h

b₁ and b₂ are the two parallel bases. h = height (perpendicular distance between them). Add the bases, multiply by the height, then divide by 2.

SECTION 5

Shape-by-Shape Guide

The diagram below puts all six shapes on one page with their formulas. Use it as a quick-reference sheet when you practice.

All six shapes with their formulas. Dashed lines show the height, which is always perpendicular (at a right angle) to the base.

Shape	Perimeter Formula	Area Formula
Square	P = 4s	A = s²
Rectangle	P = 2l + 2w	A = l × w
Triangle	P = a + b + c	A = ½ × b × h
Circle	C = 2πr or πd	A = πr²
Parallelogram	P = 2a + 2b	A = b × h
Trapezoid	P = a + b₁ + c + b₂	A = ½(b₁ + b₂) × h

Summary of formulas for six common plane figures

SECTION 6

Worked Example: A Backyard Garden

Maria wants to put a fence around her rectangular garden and then fill it with soil. The garden is 12 feet long and 8 feet wide. How much fencing does she need (perimeter)? How much soil will cover the garden (area)?

Finding the Perimeter and Area of a Rectangle

1
Step 1 — Identify the given values

The length is l = 12 ft and the width is w = 8 ft.
2
Step 2 — Write the perimeter formula

P = 2l + 2w
3
Step 3 — Substitute the values

P = 2(12) + 2(8) = 24 + 16

P = 40 ft
4
Step 4 — Write the area formula

A = l × w
5
Step 5 — Substitute the values

A = 12 × 8

A = 96 ft²
6
Step 6 — Interpret the answers

Maria needs 40 feet of fencing for the border and 96 square feet of soil to cover the garden.

⚠

Watch Out!

Perimeter is in regular units (ft). Area is in square units (ft²). If you forget the "squared" part of the area unit, you may lose a point on the TACHS.

SECTION 7

Common Mistakes & How to Avoid Them

Many students mix up formulas or forget steps. The table below shows the most common errors and how to fix them.

Mistake	Why It's Wrong	Fix
Using the slanted side as the height of a triangle or parallelogram	The height must be perpendicular (at a 90° angle) to the base	Look for a dashed line with a small square at its base — that marks the true height
Forgetting to halve the triangle area	A = b × h gives you a rectangle, not a triangle	Always multiply by ½ (or divide your answer by 2)
Using diameter instead of radius in A = πr²	If d = 10, then r = 5, not 10. Using 10 gives you 4× the correct area	If given the diameter, divide by 2 first to get the radius
Writing area in regular units (cm instead of cm²)	Area measures square units. The answer needs a ² symbol	Double-check: perimeter → regular units; area → squared units
Confusing perimeter and area formulas	Adding when you should multiply (or vice versa)	Remember: Perimeter = add sides; Area = multiply dimensions

Top 5 mistakes on perimeter and area problems

✦

KEY TAKEAWAY

Think of perimeter like wrapping a ribbon around a gift box — you are measuring the outside edge. Think of area like wrapping paper that covers the top of the box — you are measuring the flat surface. Ribbon goes around (add). Paper covers the surface (multiply).

SECTION 8

Connecting to Advanced Topics

Once you master perimeter and area of flat shapes, you are ready for bigger ideas. Here is a preview of what comes next.

What You Know Now	What Comes Next
Perimeter of flat shapes	Surface area of 3D shapes (boxes, cylinders, spheres)
Area of flat shapes	Volume of 3D shapes (how much space is inside)
Using π for circles	Volume of cylinders, cones, and spheres
Simple shapes	Composite figures (shapes made of two or more simple shapes combined)

From 2D to 3D: what's ahead

On the TACHS, you might see a problem that combines shapes. For example, you could be asked to find the area of an L-shaped room. The trick is to break it into two rectangles, find each area, and add them together. The formulas you learned today are the building blocks for those problems.

💡

Composite Figures Tip

If a shape looks unusual, try splitting it into shapes you already know (rectangles, triangles, or semicircles). Find each area separately, then add or subtract.

SECTION 9

Practice Problems

Try these five problems. They get harder as you go. Check your work against the answers below each problem.

Question 1: CONCEPTUAL

A square has a perimeter of 36 cm. What is the length of one side? Explain how you know.

Show answer

A square has 4 equal sides, so you divide the perimeter by 4. Since P = 4s, we get s = 36 ÷ 4 = 9 cm. Each side is 9 cm long. This works because perimeter is the total distance around the shape, and all four sides of a square are the same length.

Question 2: BASIC CALCULATION

Find the area of a rectangle with a length of 15 inches and a width of 9 inches.

Show answer

Use the formula A = l × w. Substitute the values: A = 15 × 9 = 135 in². Remember that area is always in square units. You are finding how many 1-inch-by-1-inch squares fit inside the rectangle.

Question 3: INTERMEDIATE

A triangle has a base of 14 cm and a height of 10 cm. Its three sides measure 14 cm, 11 cm, and 13 cm. Find both the perimeter and the area of the triangle.

Show answer

Perimeter: P = 14 + 11 + 13 = 38 cm. Area: A = ½ × b × h = ½ × 14 × 10 = ½ × 140 = 70 cm². For the perimeter, we added all three sides. For the area, we used the base-and-height formula and divided by 2. The height must be perpendicular to the base.

Question 4: APPLIED

A circular swimming pool has a diameter of 20 feet. The owner wants to build a fence around it and also buy a cover for the top. Find the circumference (for the fence) and the area (for the cover). Use π ≈ 3.14.

Show answer

First, find the radius: r = d ÷ 2 = 20 ÷ 2 = 10 ft. Circumference: C = 2πr = 2 × 3.14 × 10 = 62.8 ft. Area: A = πr² = 3.14 × 10² = 3.14 × 100 = 314 ft². The owner needs about 62.8 feet of fencing and a cover that is 314 square feet. Always convert the diameter to the radius before using these formulas.

Question 5: CRITICAL THINKING

A trapezoid has parallel bases of 6 m and 10 m, and a height of 4 m. A rectangle has the same area as this trapezoid. If the rectangle's width is 4 m, what is its length?

Show answer

First find the trapezoid's area: A = ½ × (b₁ + b₂) × h = ½ × (6 + 10) × 4 = ½ × 16 × 4 = ½ × 64 = 32 m². Now use the rectangle area formula: A = l × w, so 32 = l × 4. Divide both sides by 4: l = 32 ÷ 4 = 8 m. This problem shows that different shapes can have the same area. You needed to use two formulas and work backward to find a missing dimension.

SUMMARY

Lesson Summary

Summary

Perimeter is the distance around a shape, found by adding all the side lengths. For a rectangle, use P = 2l + 2w. For a square, use P = 4s. For a circle, the perimeter is called the circumference: C = 2πr. Perimeter is measured in regular units like cm, ft, or m.

Area is the space inside a shape, found by multiplying dimensions. A rectangle's area is A = l × w. A triangle's area is A = ½ × b × h. A circle's area is A = πr². For a trapezoid, use A = ½(b₁ + b₂) × h. Area is always in square units like cm², ft², or m². Always identify your shape, pick the right formula, plug in the numbers, and label your answer with the correct unit.