Rectangles
Find the lengths of two consecutive sides of the rectangle. Because rectangles have two pairs of sides of equal length, label one side as the base (b) and one side as the height (h). Generally, the horizontal side is the base and the vertical side is the height.
Multiply base times height to get the area. If the area of the rectangle is k, k=b*h. This means that the area is simply the product of the base and the height. For more extensive instructions, check out How to Find the Area of a Quadrilateral
Squares
Find the length of a side of the square. Because squares have four equal sides, all of the sides should have this same measurement.
Square the length of the side. This is your area. This works because a square is simply a special rectangle that has equal width and length. So, in solving k=b*h, b and h are both the same value. So, you end up squaring a single number in order to find the area.
Parallelograms
Choose one side to be the base of the parallelogram. Find the length of this base.
Draw a perpendicular line to this base, and determine the length of this line between where it crosses the base and the side opposite to the base. This length is the height. If the side opposite to the base is not long enough that the perpendicular line crosses it, extend the side along the line until it intersects the perpendicular line.
Plug the base and height into the equation k=b*h. For more extensive instructions, check out How to Find the Area of a Parallelogram
Trapezoids
Find the lengths of the two parallel sides. Assign these values to variables a and b.
Find the height. Draw a perpendicular line that crosses both parallel sides, and the length of the line segment on this line connecting the two sides is the height of the parallelogram (h).
Plug these values into the formula A=0.5(a+b)h. For more extensive instructions, check out How to Calculate the Area of a Trapezoid
Triangles
Find the base and height of the triangle. This is the length of one side of the triangle (the base), and the length of the line segment perpendicular to the base connecting the base to the opposite vertex of the triangle.
To find the area, plug the base and height values into the equation A=0.5b*h. For more extensive instructions, check out How to Calculate the Area of a Triangle
Regular Polygons
Find the length of a side and the length of the apothem (the line segment perpendicular to a side connecting the middle of a side to the center. The length of the apothem will be assigned the variable a.
Multiply the length of the side by the number of sides to get the perimeter of the polygon (p).
Plug these values into the equation A=0.5a*p For more extensive instructions, check out How to Find the Area of Regular Polygons
Circles
Find the radius of the circle (r). This is a line segment connecting the center to a point on the circle. By definition, this value is the same no matter what point you pick on the circle.
Plug the radius into the equation A=πr^2. For more extensive instructions, check out How to Calculate the Area of a Circle
Surface Area of a Pyramid
Find the area of the base rectangle by using the formula shown above for finding the area of a rectangle: k=b*h
Find the area of each side triangle by using the formula shown above for finding the area of a triangle:A=0.5b*h.
Add up all the areas: the base and all the sides.
Surface Area of a Cylinder
Find the radius of one of the base circles.
Find the height of the cylinder
Find the area of the bases using the formula of the area of a circle: A=πr^2.
Find the area of the side by multiplying the height of the cylinder by the perimeter of the base. The perimeter of a circle is P=2πr, so the area of the side is A=2πhr.
Add up all the areas: the two identical circular bases and the side. So, the surface area should be SA=2πr^2+2πhr. For more extensive instructions, check out How to Find the Surface Area of Cylinders
The Area Under a Function
Define f(x) in terms of x.
Take the integral of f(x) within [a,b]. By the Fundamental Theorem of Calculus, given F(x)=∫f(x), ∫abf(x) = F(b)—F(a).
Plug in the a and b values into the integral expression. The area under f(x) between x [a,b] is defined as ∫abf(x). So, A=F(b))—F(a).
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