ASVAB · Mathematics Knowledge · 6 min read
Perimeter, Area, Circles and Volume
- Calculate the perimeter and area of common two-dimensional polygons and composite figures without using a calculator.
- Solve problems involving circles, semicircles, circumference, and circular area using radius and diameter relationships.
- Determine the volume, surface area, and lateral area of three-dimensional prisms and cylinders using appropriate units.
Linear Measurement: Perimeter of Polygons
Perimeter is the total linear distance around the boundary of a two-dimensional shape. For any polygon, you find the perimeter simply by adding the lengths of all its outer sides. If you were building a fence around a yard or running border trim along a wall, you would be measuring perimeter.
Certain polygons have equal sides or matching pairs of sides, which gives rise to useful shortcuts. For a rectangle with length l and width w, the perimeter formula is P = 2l + 2w. For a square where all four side lengths are s, the formula simplifies to P = 4s. Similarly, an equilateral triangle has three sides of equal length, giving it a perimeter of P = 3s. For any regular polygon with n sides of equal length s, the perimeter is calculated as P = n * s.
Always pay attention to the units of measure before calculating. If dimensions use different units, convert them to the same unit before calculating perimeter, area, surface area, or volume. For instance, if one side of a shape is given in feet and another in inches, you must convert the measurements to a matching unit before doing any addition.
Common mistakes
- Plugging the diameter directly into the circular area formula without dividing by two to find the radius first.
- Using a slanted side length instead of the perpendicular height when calculating the area of a triangle or parallelogram.
- Expressing volume in square units or surface area in cubic units instead of using cubic units for volume and square units for surface area.
- Failing to convert mixed measurements to matching units (such as combining feet and inches) before performing perimeter, area, or volume calculations.
Key points
- Perimeter is linear distance around a polygon found by adding outer sides, while area measures flat surface space in square units.
- Height in area and volume formulas always refers to the perpendicular distance to the base.
- A circle's diameter is twice its radius (d = 2r); circumference is C = 2 * pi * r or C = pi * d, and area is A = pi * r^2.
- Doubling a circle's radius doubles its circumference but quadruples its area.
- Right prism volume equals base area times height (V = B * h), cube volume is V = s^3, and cylinder volume is V = pi * r^2 * h.
- Surface area is measured in square units; a closed cylinder's total surface area is SA = 2 * pi * r^2 + 2 * pi * r * h, where the lateral area alone is 2 * pi * r * h.
- Always convert all dimensions to the same unit before performing calculations without a calculator.
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