# Topic Category: ACT (Math)

## Areas of Complex Forms: Focusing on the Building Blocks

You might see complex figures like the one in Figure 6, where thinking in terms of a single shape will not be quite enough. Let’s say all you know about the figure is that the central hexagon is a regular hexagon (all sides have the same length ‘s’) and some dimensions for what looks like …

## Areas of Circles and Sectors

Now we move on to the world of coins and pizzas. A circle’s diameter is the distance of a straight line from one point on a circle to another point on the opposite edge while passing its center. The radius (plural: radii) is half the diameter and is the basis for calculating the area of a circle.  A sector of a circle …

## Areas of Regular Polygons

A regular polygon is any polygon that has all sides equal (equilateral) and all angles equal (equiangular). Some familiar examples we have seen so far are equilateral triangles and squares. As the number of sides increase to include pentagons (5-sided), hexagons (6-sided), and even decagons (10-sided), we often call them n-gons where n is the number of sides. Therefore, regular n-gons are …

## Areas of Trapezoids

Whereas a parallelogram has two pairs of parallel sides, a trapezoid is a quadrilateral with only one pair of parallel sides. The formula for the area of any trapezoid is where and are the lengths of each base of the parallel pair. We can all agree that this formula is a little more complex than …

## Areas of Parallelograms and Triangles

The formula for the areas of parallelograms and triangles are very similar to that for rectangles. Again, all we need is the length of the base and height. First, let’s look at parallelograms. A parallelogram is any four-sided shape (quadrilateral) with two pairs of parallel sides. Looking at the parallelogram in Figure 2, you might notice …

## Areas of Rectangles and Squares

The area of any rectangle is the product of its base and height. Since squares are special rectangles with four equal sides, their areas are found as the product of two sides (See Figure 1). Once you have chosen one side as your base, keep in mind that the segment chosen as your height must be perpendicular (forms a right angle) …

## Introduction to Area

Students frequently encounter problems in calculating the area of shapes ranging from simple triangles to complicated shapes. Not many students realize, however, that they intuitively determine areas of various surfaces every day. You know right away that your iPad has a larger area than your laptop and fits in your backpack more easily. You don’t …

## Complementary Angles of the Triangle

Complementary Angles Complementary angles are two angles whose sum is . For example, if , we can say that are complementary angles. Supplementary Angles Supplementary angles are two angles whose sum is . For example, if and that , we can say that are supplementary angles. GRANITE QUICK TRICK™ A useful tip to remember the …

## Right Triangles

Right triangles are one of the most special triangles of all! We would be remiss to skip over the right triangle. The name comes from the right angle, which is an angle whose measure is precisely . Therefore, a right triangle is a triangle which has one right angle. Remember, the right triangle is still …

## Introduction to Triangles

Triangles are three-sided polygons that are formed from line segments joined at three vertices. A vertex (plural: vertices) refers to a corner or a point where lines meet (see Figure 1). However, one of the most important properties of triangles has to do with the three angles that define the fundamental properties of the polygon. Let’s see how …