What is area?
Areas refer to the space within a two-dimensional object. These include 2D shapes such as squares and triangles. In the real world, area can measure things such as the space provided by a floor or garden, or the space on a table top.
Area of Rectangles
One of the first area formulas students learn, finding the area of a rectangle is as easy as multiplying the length times the width. Sometimes, you will also see the sides of a rectangle referred to as the base (the horizontal part on the “floor”) and the height (the vertical part). While this formula is fairly simple, it’s important that students familiarize themselves with other terms for length and width, including breadth, depth, base, height, and span.
Area of Squares
Squares have the same area formula as rectangles, but all of their sides are equal. That means that you can also express the area formula for a square as the side squared. The ACT and SAT expect you to know this and make this connection. Often, problems involving the area of squares will state the area and the shape being a square, expecting you to know you can take the square root of the area to find a side length or square the side to find the area.
Area of Triangles
The area formula for triangles is 1/2bh. The formula is pretty straight forward, but there are two areas were students get fooled by these problems. The first regards finding the proper height of the triangle. Remember that the height of a shape is always a completely vertical straight line and will always be perpendicular to the base. If the line has any diagonal to it, it cannot be the height. Think of it this way: whenever you measured your height as a kid, you didn’t lean against the wall any way you pleased. You have to stand up as straight as possible against the wall. It’s the same with shapes.
The second thing that trips up students is not paying proper attention to the orientation of the triangle. When given a triangle, it will not always be right side up. Sometimes it will be drawn on its side or upside down. Always keep in mind that the height must be the longest straight line perpendicular to the base. If it’s a right triangle, the base and height will always be next to the right angle.
Area of Circles
To find the area of a circle, students must be familiar with finding the radius and the diameter of the shape. The radius extends from the center of the circle to its edge, and the diameter is twice that length, going from one end of the circle to the other through the center.
For circle problems, always pay attention to the answer choices. Sometimes pi (π) is written in the answer, meaning you can save yourself some work and not have to multiply by 3.14 or using the π button in your calculator.
Area of Parallelograms
Even though parallelograms look a little different, their area formula is the same as a rectangle’s: base x height.
Area of Trapezoids
You can find the area of a trapezoid by adding the two sides that are parallel to one another, dividing by 2, and multiplying times the height.
Dividing an Area
One type of problem might ask you to divide an area into pieces. To do this, find the area of the larger shape, the area of the smaller pieces, and divide the larger number by the smaller one. For example, if you wanted to cover the floor of a rectangular room with tile, you would need to know the dimensions of the room and the tiles. If our room measures 12 ft by 14 ft and we have tiles that are 2 ft by 2 ft, that would mean we’re dividing a 168 sq ft room by 4 sq ft tiles. That means we can fit 42 of our tiles in this room. One danger students need to be aware of in this type of problem is unit conversions. Often, the dimensions of one shape may be given in a different dimension than the smaller shapes. Make sure to convert your units BEFORE finding any of the areas or your numbers will be off.
Finding the Area of Composite Shapes
If you see an odd-looking shape on the test, don’t freak out. It’s not that difficult to find the area of these shapes too. Look carefully and divide the shape into its components. After you do that, find the area of each of the component shapes and add them together. Take the shape above for example.
This shape can be separated into a rectangle and a semi-circle. Problems like these make it easy to find the base and the height of the rectangle, but some may get stuck on solving for the semi-circle. Think about the properties of a rectangle. They’re a type of parallelogram, so their opposite sides are congruent. That means the side touching the semi-circle is equal to the outer part. Since the side goes across the length of the semi-circle, it is also the diameter. Below, you can see this shape divided. So the sum of 10 x 20 for the rectangle and π times r squared (divided by 2) for the semi-circle gives us the area for this shape (NOTE: remember that a semi-circle is half the area of that of a circle – don’t forget to divide by 2!). The final answer is 200 + 25π/2.
Finding the Shaded/Remaining Area
A shaded area problem is opposite of a composite shape problem. In this case, something is being removed from one shape, and you’re asked to find the remaining area. Like before, start by finding the area of all shapes. Then subtract the area of the inner shapes from the larger one.
One common iteration of this problem is an inscribed circle problem. In these questions, the circle is drawn inside the square so that its sides touch the sides of the square. That means that if you were to draw the diameter of the circle, it would reach from one end to the other of the square, parallel to its sides! If the above circle has a radius of 4, that means the diameter of the circle (and also the side of the square) is 8. The area of the square is then 64, and the area of the circle is 16π. The shaded area is 64 – 16π.
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