Volume and Area on the SAT
What is the difference between perimeter and area?
- Perimeter is the length of the boundary of a 2-dimensional (flat) figure. E.g. the perimeter of a square is the sum of all its 4 sides
- Area is the size of a surface i.e. the amount of space inside the boundary of a flat (2-dimensional) object. Area of a rectangle is the product of its length and breadth. Since area is derived by multiplying two dimensions, it is expressed in square units.
- Figures with the same area can have different perimeters; and figures with the same perimeter can have different areas.
Let us illustrate with an example:
All the three figures above have an area of 9 but different perimeters:
- Figure 1 is 20
- Figure 2 is 12
- Figure 3 is 14
What is the difference between surface area and volume?
- Surface area is the sum of the areas of all the faces of the solid figure. It is measured in square units.
- Volume is the interior of a 3-dimensional object (solid figure). It is measured in cubic units.
An easy way to remember the formulas for finding area and volume:
- Area is measured in square units and volume is measured in cubic units.
- Square units are units x units
- Cubic units are units x units x units
- e.g. In the formula for the area of a rectangle, A = l x w, you multiply two dimensions. In the formula for the volume of a rectangular prism, V = l x w x h, you multiply three dimensions.
Perimeter and Area
In general, the perimeter of a shape is the sum of the lengths of its sides. A polygon is a closed shape made up of several straight line segments. The perimeter of a polygon is the sum of the length of all these sides.
For example, the perimeter of the polygon shown below is 1 + 5 + 4 + 2 + 7 = 19
Surface Area and Volume of simple objects
The surface area can be found simply by summing up the area of all the surfaces of the object.
The volume of most solids can be found by multiplying the area of the base times the height.
A rectangular solid is one kind of prism. You may occasionally run into another shape of prism such as a triangular or hexagonal prism. A prism is a solid in which the top and bottom are a shape (like a triangle), and the sides are rectangles.
All prisms have the same formula for volume: area of the base times the height.
Some other formulae
- Long Diagonal of a Rectangular Solid (Super Pythagorean Theorem)
- Face Diagonal of a Cube f = s√2
Long Diagonal of a Cube d = s√3
The longest line that can be drawn inside a cylinder is the diagonal of the rectangle formed by the diameter and the height of the cylinder. You can find its length with the Pythagorean theorem.
Cones, Pyramids and Spheres
Cones, Pyramids and Spheres don’t show up that often on AAT, and when they do they most likely won’t be testing you on volume. You may be tested on surface area, or more often, ratios of the top to the bottom or something like that. If they are testing you on volume or anything more obscure, they will give you the formula in the body of the SAT question.
where r is the radius, h is the height, and l is the slant height. It is important not to confuse the slant height with the height. The height is used to find volume. The slant height is used to find surface area.
Volume of right circular cone with radius r and height h: V = 1/3Πr²h
Lateral area of cone with base radius r and slant height l: S = Πrl
Total surface area of cone = Area of base + lateral area of cone = Πr² + Πrl
The base of a pyramid is a rectangle.
Volume of pyramid with base area B and height h: V = 1/3Bh where B = wl
In both cones and pyramids, if you draw a line parallel to the base that divides a top portion from the lower portion, the smaller cone or pyramid formed by that line will be similar to the original cone or pyramid. This is most often what will be tested.
Like a circle, all radii in a sphere are equal.
Volume of sphere with radius r: V = 4/3Πr³
Surface Area of sphere with radius r: S = 4Πr²