Math Study Guide for the SAT Exam

Geometry: Shapes

In geometry, we deal with two-dimensional and three-dimensional figures. Knowing how to measure and manipulate both types of figures is a foundational skill in geometry.

Area and Volume Formulas

One way to study the various shapes is to measure their area and volume. Knowing the formula to use is vital for accurate measurement.

Area Formulas for Two-Dimensional Shapes

Any closed two-dimensional figure will possess an area. The area of a figure is simply the amount of space enclosed by the figure. Because areas are two dimensional, each area measurement will have units raised to the second power (i.e., units squared). You should commit the following area formulas to memory:

• area of a circle—\(\pi r^2\), where \(\pi\) is a universal constant (\(\approx 3.1459\)) and \(r\) is the radius of the circle.

• area of a triangle— \(\frac{1}{2}b h\), where \(b\) is the base and \(h\) is the height

• area of a rectangle—\(l w\), where \(l\) is the length (or base) and \(w\) is the width (or height)

• area of trapezoid— \(\frac{1}{2} (b_1 + b_2) h\), where \(b_1\) and \(b_2\) are the two bases and \(h\) is the height

Area of Composite Shapes

There will be questions that ask you to find the area of a figure composed of many different shapes. These are called composite or combined shapes. To do this, you simply break the figure apart into its constituent shapes, calculate the area of each of the shapes, then add each of the areas together to find the total area. Consider the figure shown below:

What is the area, in square units, of the composite figure?

Solution

First, we note that the composite figure is composed of a triangle and a semicircle. We will find the areas of each of these shapes separately and then add them to find the area of the whole figure.

The area of a circle is found using the formula \(A = \pi r^2\), where \(r\) is the radius, so the area of a semicircle (half a circle) can be found using the formula \(A = \frac{1}{2} \pi r^2\).

Note: We aren’t told the units for the given measures, so we can ignore them in our calculations, but just remember that they would be square units because we’re finding area.

From the above figure, we can find:

\[A = \frac{1}{2} \pi (5)^2 = \frac{25}{2} \pi\]

Now, the area of a triangle can be found using the formula \(A = \frac{1}{2} bh\), where \(b\) is the base and \(h\) is the height. From the figure, we see that twice the radius is the base of triangle, so \(b = 5 \times 2 = 10\) and the height, \(h\), is \(12\). So, the area of the triangle is:

\[A = \frac{1}{2} (10)(12) = 60\]

Thus, the area of the composite figure is \(\frac{25}{2} \pi + 60\) (square units).

Volume Formulas for Three-Dimensional Shapes

Three-dimensional figures possess both a surface area and a volume. The surface area is the sum of the areas of each of the figure’s surfaces. In order to find the surface area of a figure, first, calculate the area of each surface composing the figure and then add the areas together.

The volume of a three-dimensional figure (or object) is the amount of space enclosed by the figure. In the same way that a shape’s area is composed of two dimensions and represented by units squared, an object’s volume is composed of three dimensions and represented by units cubed.

You should commit the following volume formulas to memory:

• volume of a sphere— \(\frac{4}{3} \pi r^3\), where \(r\) is the radius of the sphere

• volume of a cylinder—\(\pi r^2 h\), where \(r\) is the radius of the base, and \(h\) is the cylinder’s height

• volume of a rectangular prism—\(l w h\), where \(l\) is the length, \(w\) is the width, and \(h\) is the height

Triangles

Triangles are polygons with three vertices at the intersection of three line segments. The interior angles of a triangle add to \(180\) degrees.

These are common types of triangles you should know:

• Isosceles triangles are triangles with two side lengths of equal measurement and two angles of equal measurement.

• Equilateral triangles have three side lengths of equal measurement. Equilateral triangles also contain three \(60\)-degree angles.

• Right triangles contain two acute angles and a right angle (\(90\) degrees). Their side lengths can be related through the Pythagorean theorem (discussed below). There are two special right triangles you should be familiar with:

  • A \(\bf{30-60-90}\) triangle has angle measurements of \(30\), \(60\), and \(90\) degrees. The side lengths of these triangles are always in the ratio of \(x\), \(x\sqrt{3}\), and \(\,2x\), corresponding to the \(30-60-90\) angles.

  • A \(\bf{45-45-90}\) triangle has angle measurements of \(45\), \(45\), and \(90\) degrees. The side lengths of these triangles are always in the ratio of \(x\), \(x\), and \(x\sqrt{2}\), corresponding to the \(45-45-90\) angles.

Triangle Side Lengths

In order to verify that a proposed triangle is valid, compare the sum of two side lengths with the length of the third side. Any two sides of a valid triangle, when added together, must always be longer than the third side.

Consider the following example: Is a three-sided polygon with side lengths \(4\), \(4\), \(12\) a valid triangle?

Compare the sum of each two side lengths to the third length to determine whether a valid triangle can be made:

\[4 + 4 > 12\] \[8 > 12\]

This is not true, so it is impossible to have a triangle with the given side lengths.

The Pythagorean Theorem

The Pythagorean theorem relates the three sides of a right triangle.

The shorter sides of a right triangle are known as the legs, and the longest side is known as the hypotenuse. The Pythagorean theorem states that the sum of the squares of the legs (\(a\) and \(b\)) is equal to the square of the hypotenuse (\(c\)):

\[a^2 + b^2 = c^2\]

Other Polygons

• Regular polygons are polygons with angles and side lengths that are all equal in measure.

• Squares, which are quadrilaterals, are regular polygons because each of the angles measures \(90\) degrees and all side lengths measure the same length. The perimeter of a square is equal to \(4s\) where \(s\) is the side length. The area of a square is equal to \(s^2\). Note: A square is a type of rectangle.

• A parallelogram is a quadrilateral with parallel lines composing opposite sides. The properties governing transversals and parallel lines also apply to parallelograms. The perimeter of a parallelogram is found by summing the length of each side. The area of a parallelogram is equal to \(bh\), where \(b\) is the base and \(h\) is the height. Note: Parallelograms include squares and rectangles.

• A trapezoid is a quadrilateral composed of two parallel lines. Its perimeter is found by summing each of its sides. Its area is found by multiplying the average of its bases by its height.

• A rhombus is a quadrilateral that contains equal side lengths but two pairs of different angle measurements. Its perimeter is the sum of its sides and its area is half of the product of its diagonals. In other words, if a rhombus has diagonals \(p\) and \(q\), then the area of the rhombus is:

\[A = \frac{1}{2} pq\]

Congruency and Similarity of Polygons

Geometric figures sharing the same attributes (e.g.,line segments of the same length, polygons with the same angle measurements) are said to be congruent. Congruency is designated with this symbol: \(\cong\).

Figures and shapes that are not the same size but have proportional measurements are said to be similar. Similarity is designated with this symbol: \(\sim\).