Geometry Basics: Area and Pythagoras

1.Every Area Formula Comes from the Rectangle

Area is a count of how many squares with sides of 1 fit inside a shape. A room 6 m wide and 5 m long holds 30 tiles of 1 m × 1 m, so its area is 30 m². The area of a rectangle = width × length is the starting point for every area formula.

If you cut the triangle off one end of a parallelogram and attach it to the other side, you get a rectangle, so its area is base × height. Two identical triangles put together make a parallelogram, so a triangle is half of that: base × height ÷ 2. Putting two identical trapezoids together, one upside down, makes a parallelogram whose base is (top side + bottom side), so a trapezoid is half of that. Be sure to remember that the height here is not the length of a slanted side but the distance perpendicular to the base.

2.Circles: Circumference and Area

If you divide a circle's circumference by its diameter, you always get the same value, no matter how big the circle is. This value is pi (π), about 3.14. π is a decimal that goes on forever without any repeating pattern of digits, so in calculations you either approximate it as 3.14 or leave the answer with π in it.

You can see why a circle's area is π × r² by cutting the circle into thin slices like a pizza and laying them out alternately. The more slices there are, the closer the shape gets to a rectangle whose height is the radius r and whose width is half the circumference, πr, so the area is πr × r = πr².

Area is proportional to the square of length. If the radius becomes 2 times as long, the area becomes 4 times as large; if it becomes 3 times as long, 9 times. This sense is very useful when comparing pizzas, flowerpots, and screen sizes.

3.Similarity and Scale: How Length, Area, and Volume Grow

Two shapes with the same form but different sizes are called similar. When lengths in similar shapes are multiplied by k, areas are multiplied by k² and volumes by k³. That's because area is a product of two lengths and volume is a product of three. If you stretch the edges of a cube from 1 cm to 2 cm, its surface area becomes 4 times as large, and its volume goes from 1 cm³ to 8 cm³, 8 times as large.

Map scales work on the same principle. A scale of 1 : 50,000 means 1 cm on the map is 50,000 cm, or 500 m, in reality. For lengths, you just multiply by 50,000, but for areas you must multiply by 50,000², not 50,000. Forgetting this and multiplying only by the length ratio is the most common mistake. If you get confused, the safe way is not to convert the area directly, but to convert each side to its real length first and then calculate the area.

4.The Pythagorean Theorem

In a right triangle, if the two sides that form the right angle are a and b, and the longest side, opposite the right angle (the hypotenuse), is c, then a² + b² = c². This means that if you know the lengths of two sides, you can find the third.

3, 4, 5 is the most famous set of whole numbers that satisfies this relationship: 9 + 16 = 25. Its scaled-up version 6, 8, 10, and the set 5, 12, 13 (25 + 144 = 169), also come up often. Conversely, if the three sides of a triangle satisfy this relationship, it is a right triangle. That's why carpenters measure 3 m and 4 m and check whether the diagonal is 5 m to test whether a corner is square.

5.The Distance Between Two Points on a Coordinate Plane

The distance between two points on a coordinate plane is also found with the Pythagorean theorem. Draw a right triangle whose hypotenuse is the segment joining the two points and whose other two sides are the horizontal difference and the vertical difference.

This distance is used just the same way when working with data. If you describe two people's tastes with two numbers each, the closer the two points are, the more similar their tastes can be considered. Even when the numbers grow to three or a hundred, the method stays the same: "square the differences, add them all up, and take the square root." The vectors and similarity in Lesson 10 carry this idea further.