Use properties of basic geometric shapes (triangles, rectangles, circles) - TACHS Math

SECTION 1

Where Did Geometry Come From?

People have used shapes for thousands of years. Ancient farmers needed to measure their land after floods washed away boundary markers. Builders needed to make walls straight and corners square. The word geometry actually comes from two Greek words meaning "earth" and "measure." So geometry literally means "measuring the earth!"

Over time, people discovered rules about shapes that always work. These rules, called properties, help us calculate things like how much fencing you need around a yard or how much pizza is in a large pie. Let's look at how this knowledge developed.

3000 BCE • Ancient Egypt

Egyptian surveyors used ropes and stakes to re-draw field boundaries after Nile River floods. They discovered the 3-4-5 triangle trick to make perfect right angles.

2000 BCE • Babylonian Tablets

Babylonians carved clay tablets showing area formulas for rectangles and triangles. They even estimated the value of pi (π) for circles.

300 BCE • Euclid's Elements

The Greek mathematician Euclid wrote a famous textbook called "Elements." It organized all known geometry rules into a logical system still used today.

Today • Modern Applications

Architects, engineers, video game designers, and even doctors use properties of basic shapes every day to build, create, and solve problems.

So here's the big question: what are the specific properties of triangles, rectangles, and circles that make them so useful? Let's find out!

SECTION 2

Core Properties You Need to Know

Every shape has special features that never change, no matter how big or small you draw it. These features are called properties. Once you know a shape's properties, you can find missing measurements like side lengths, angles, and area.

Triangle Properties

A triangle has 3 sides and 3 angles. The angles always add up to exactly 180°. You find its area with ½ × base × height.

Rectangle Properties

A rectangle has 4 sides with 4 right angles (each 90°). Opposite sides are always equal in length. Area = length × width.

Circle Properties

A circle has no sides or angles. Every point on the circle is the same distance from the center. That distance is called the radius.

Perimeter & Circumference

The distance around a triangle or rectangle is its perimeter (add up all sides). For a circle, this distance is called the circumference.

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KEY TAKEAWAY

Think of shape properties like the rules of a board game. A triangle's angles must add to 180° just like a basketball game has four quarters. These rules never change, so once you memorize them, you can always figure out the missing pieces!

SECTION 3

Seeing the Three Shapes

A picture is worth a thousand words! The diagram below shows a triangle, a rectangle, and a circle side by side. Pay attention to the labeled parts — these are the measurements you'll use in formulas.

The triangle shows a base and height used in area calculations. The rectangle shows its length, width, and 90° corners. The circle highlights the center, radius, and diameter.

Notice that the triangle's height is always measured straight up from the base (at a right angle). The rectangle's four corners are all 90° right angles. And the circle's radius goes from the center to any point on the edge.

SECTION 4

Key Formulas for Each Shape

Now let's look at the formulas you'll use on the TACHS. Don't worry — each one is pretty short. The trick is knowing which formula goes with which shape.

Triangle Formulas

TRIANGLE AREA

A = ½ × b × h

A = area, b = base (any side), h = height (measured at a right angle to the base)

TRIANGLE ANGLE SUM

Angle₁ + Angle₂ + Angle₃ = 180°

The three interior angles of any triangle always add up to 180 degrees. If you know two angles, subtract their sum from 180° to find the third.

Rectangle Formulas

RECTANGLE AREA

A = l × w

A = area, l = length, w = width. Simply multiply the two side lengths.

RECTANGLE PERIMETER

P = 2l + 2w

P = perimeter (total distance around). Add the length twice and the width twice because opposite sides are equal.

Circle Formulas

CIRCLE AREA

A = π × r²

A = area, π ≈ 3.14, r = radius. Square the radius first, then multiply by π.

CIRCUMFERENCE

C = 2 × π × r (or C = π × d)

C = circumference (distance around the circle), d = diameter (the full width through the center). Remember: d = 2r.

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Quick Tip

On the TACHS, you can usually use π ≈ 3.14. If the answer choices are written with π (like 25π), just leave π in your answer — don't multiply it out.

SECTION 5

Types of Triangles and Special Rectangles

Not all triangles look the same! They are classified (sorted into groups) by their sides and their angles. Knowing the type of triangle can help you spot shortcuts. Rectangles also have a special cousin called the square.

Top row: Four types of triangles classified by sides and angles. Bottom row: A rectangle has opposite sides equal, while a square (a special rectangle) has all four sides equal.

Triangle Type	Sides	Angles
Equilateral	All 3 sides equal	All 3 angles = 60°
Isosceles	2 sides equal	2 base angles equal
Scalene	No sides equal	No angles equal
Right	Longest side is hypotenuse	One angle = 90°

Summary of triangle classifications

SECTION 6

Step-by-Step Worked Example

Let's solve a problem that uses properties of all three shapes. Read the problem carefully, then follow each step.

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Problem

A park has a rectangular soccer field that is 80 meters long and 50 meters wide. In one corner, there is a triangular flower bed with a base of 10 m and a height of 6 m. In the center of the field is a circular fountain with a radius of 4 m. What is the area of the field that is just grass (not the flower bed or fountain)? Use π ≈ 3.14.

Finding the Grass Area

1
Step 1 — Find the rectangle's area

Use A = l × w. Plug in: A = 80 × 50.

Rectangle area = 4,000 m²
2
Step 2 — Find the triangle's area

Use A = ½ × b × h. Plug in: A = ½ × 10 × 6 = ½ × 60.

Triangle area = 30 m²
3
Step 3 — Find the circle's area

Use A = π × r². Plug in: A = 3.14 × 4² = 3.14 × 16.

Circle area = 50.24 m²
4
Step 4 — Subtract the non-grass areas

Grass area = Rectangle area − Triangle area − Circle area. So, Grass = 4,000 − 30 − 50.24.

Grass area = 3,919.76 m²

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STRATEGY TIP

When a problem asks for a leftover area, first find the total area, then subtract the parts you don't want. It's like figuring out how much pizza is left: start with the whole pie and subtract the slices that were eaten.

SECTION 7

Comparing the Three Shapes

Each shape has strengths and common mistakes to watch for. The table below compares key features so you can see how they're alike and different.

Feature	Triangle	Rectangle	Circle
Number of sides	3	4	0 (curved)
Angle sum	180°	360° (4 × 90°)	No angles
Area formula	½ × b × h	l × w	π × r²
Perimeter / Circumference	Add all 3 sides	2l + 2w	2πr or πd
Common mistake	Forgetting the ½ in area	Mixing up area and perimeter	Using diameter instead of radius in πr²

Shape comparison chart

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WATCH OUT!

The most common test mistake is confusing the radius and diameter of a circle. The diameter is the full distance across the circle. The radius is half that. If a problem gives you the diameter, divide by 2 before plugging into πr². Think of it this way: the radius is like one arm reaching from your body (the center) to your fingertip (the edge). The diameter is both arms stretched out.

SECTION 8

Connecting to More Advanced Geometry

The properties you're learning now are the building blocks for everything you'll study later in geometry and even in high school math. Here's a sneak peek at how these ideas grow.

What You Know Now	What Comes Next
Triangle angle sum = 180°	Any polygon's angle sum = (n − 2) × 180°, where n is the number of sides
Area of a triangle = ½ × b × h	Heron's formula lets you find area using just the three side lengths
Rectangle area = l × w	Surface area and volume of 3D boxes (rectangular prisms)
Circle area = πr²	Volume of cylinders (πr²h) and spheres (⁴⁄₃πr³)

From basic shapes to advanced geometry

You don't need to memorize the advanced formulas right now. The important thing is that the same thinking — identify the shape, pick the right formula, plug in the numbers — works at every level. Master it now, and harder problems will feel much more manageable later.

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. After each one, check the answer to see if you're on the right track.

Question 1: CONCEPTUAL

A triangle has two angles that measure 45° and 90°. What is the measure of the third angle?

Show answer

The three angles of a triangle always add up to 180°. Add the two known angles: 45° + 90° = 135°. Subtract from 180°: 180° − 135° = 45°. This is a right isosceles triangle because it has a 90° angle and two equal 45° angles.

Question 2: BASIC CALCULATION

Find the area of a rectangle with a length of 12 cm and a width of 7 cm.

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Use the rectangle area formula: A = l × w. Plug in the values: A = 12 × 7 = 84 cm². Remember that area is always measured in square units (cm²). The perimeter would be 2(12) + 2(7) = 38 cm, which is different from area.

Question 3: INTERMEDIATE

A circle has a diameter of 10 inches. What is its circumference? Use π ≈ 3.14.

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First, find the radius: r = d ÷ 2 = 10 ÷ 2 = 5 inches. Now use C = 2 × π × r = 2 × 3.14 × 5 = 31.4 inches. You could also use C = π × d = 3.14 × 10 = 31.4 inches. Both formulas give the same answer. The key step is knowing whether you were given the radius or the diameter.

Question 4: APPLIED

Maria wants to paint one wall of her room. The wall is a rectangle that is 10 feet tall and 14 feet wide. There is a triangular window on the wall with a base of 4 feet and a height of 3 feet. She will not paint the window. How many square feet does she need to paint?

Show answer

Step 1: Find the wall area. A = l × w = 10 × 14 = 140 ft². Step 2: Find the window area. A = ½ × b × h = ½ × 4 × 3 = 6 ft². Step 3: Subtract the window from the wall: 140 − 6 = 134 ft². Maria needs to paint 134 square feet. This is a great example of subtracting a shape you don't need from the larger shape.

Question 5: CRITICAL THINKING

An equilateral triangle and a square both have the same perimeter of 24 cm. Which shape has the greater area?

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Equilateral triangle: Each side = 24 ÷ 3 = 8 cm. The height of an equilateral triangle with side 8 is about 6.93 cm (you can estimate this). Area = ½ × 8 × 6.93 ≈ 27.7 cm². Square: Each side = 24 ÷ 4 = 6 cm. Area = 6 × 6 = 36 cm². The square has the greater area (36 cm²). This shows that shapes with the same perimeter can have very different areas. In general, rounder and more compact shapes hold more area for the same perimeter.

SUMMARY

Lesson Summary

Summary

In this lesson, you learned the key properties of three basic geometric shapes. A triangle has 3 sides and 3 angles that always add up to 180°, and its area is ½ × base × height. A rectangle has 4 right angles, opposite sides that are equal, area = length × width, and perimeter = 2l + 2w. A circle has no sides or angles, and every point on it is the same distance (the radius) from the center. Its area is πr² and its circumference is 2πr.

You also learned to classify triangles as equilateral, isosceles, scalene, or right. For test success, always identify the shape first, choose the correct formula, plug in your values, and check your units. Watch out for the common trap of confusing radius and diameter in circle problems!