Area, perimeter, surface area, and volume Math 124

Perimeter Perimeter is the distance around a two- dimensional figure. We assume the figure is closed. A special case is when the figure is a polygon: then the perimeter is equal to the sum of the lengths of all the sides.

Area Area is the amount of space a two-dimensional region occupies. The figure again needs to be closed. The word area can refer both to an attribute and measurement. Area of a region is the number of area units (typically squares) required to cover the region. In the case of a rectangle with sides with integer side lengths, say l and w, the rectangle can be covered by a grid of w rows and l columns, for a total of lw unit squares. This formula can be generalized to any rectangle.

Perimeter and area Perimeter and area are not related. Bigger area may mean bigger, smaller, or equal perimeter. Of all rectangles with a fixed perimeter, however, a square has the largest area. Of all two-dimensional figures with the same perimeter, a circle has the largest area. Of all rectangles with the same area, the square has the smallest perimeter.

Area formulas Rectangle A = lw Square A = l 2 Parallelogram A = lh Trapezoid A = (b 1 +b 2 )/2*h Kite A = d 1 *d 2 /2 Triangle A = lh/2 Circle A = πr 2 Other polygons don’t have special area formulas. Areas can always be found by breaking up the polygon into triangles. Areas of irregular figures are found using calculus. Section 25.1 goes over these formulas in more detail.

Area of a kite Show that the formula for the area of a kite is Hint: What is special about the diagonals of a kite?

Volume Volume is the amount of space a three-dimensional solid occupies. It is the number of unit cubes that fit into the solid. A two-dimensional figure does not have volume. Volume in three dimensions corresponds to area in two dimensions. Using the same type of reasoning as for rectangle, a rectangular box has volume V = lwh. Look at Discussion 3 on page 583.

Surface area Surface area of a 3D shape is the number of area units needed to cover the shape. It is analogous to perimeter for two-dimensional figures. In the special case of polyhedra, the surface area is equal to the sum of the areas of all the faces of the polyhedron in question. We would be interested in surface area in real-life applications when looking for the amount of packaging of a product, for example.

Cones and cylinders A sphere is the set of all points in space at an equal distance from a fixed point (called the center of the sphere). Definition of a cylinder: It is a solid obtained by tracing around a given planar curve with a line, always keeping the line parallel to its earlier positions, and then cutting the resulting infinite surface with two parallel planes. A special case is when the planar curve is a circle, and this is the cylinder that we usually refer to. Definition of a cone: It is a solid obtained by tracing around a given planar curve with a line, always having the line go through a fixed point, and then cutting the resulting infinite surface with a plane. If the original curve is a circle, we get the cone we are used to.

Convincing yourself that the volume formulas work We will start with Discussion 3 from Section to show that the volume of a prism and cylinder can be computed using the formula Next, each group will be given a prism and a pyramid, or a cylinder and cone or hemisphere. What do you notice about their bases?

Fill the pyramid/cone/hemisphere with water and transfer this water to the prism/cylinder. How many times do you need to transfer the water from the pyramid to the prism? What does this tell you about the volume of the pyramid/cone/sphere?

Volume formulas Rectangular box V = lwh Cube V = l 3 Prism V = BH Pyramid V = BH/3 Sphere V = 4/3 π r 3 Circular cylinder V = π r 2 h Circular cone V = 1/3 π r 2 h

Surface area formulas Rectangular box SA = 2lw + 2wh + 2lh Cube SA = 6l 2 Sphere SA = 4π r 2 Circular cylinder SA = 2 π r 2 + 2π rh Others are computed using area formulas.