Solids That Aren’t Prisms
Solids That Aren’t Prisms
The solids in this section do not have two congruent bases that lie in parallel planes and thus cannot be considered prisms. Like prisms, however, the two most important aspects you’ll need to know for the test are volume and surface area.
Cones
A cone is not a prism, but it is similar to a cylinder. A cone is essentially a cylinder in which one of the bases is collapsed into a single point at the center of the base.
The radius of a cone is the radius of its one circular base. The height of a cone is the distance from the center of the base to the apex (the point on top). The lateral height, or slant height, of a cone is the distance from a point on the edge of the base to the apex. In the previous figure, these three measurements are denoted by r, h, and l, respectively.
Notice also that the height, radius, and lateral height of a cone make up a right triangle. This means that if you know the value for any two of these measurements, you will always be able to find the third by using the Pythagorean theorem.
Volume of a Cone
Since a cone is similar to a cylinder, except collapsed to a single point at one end, the formula for the volume of a cone is a fraction of the formula for the volume of a cylinder:
where r is the radius and h is the height.
For practice, find the volume of the cone pictured below:
To answer this question, just use the formula for the volume of a cone with the following values plugged in: r = x, l = 2x, and h = x. The volume is 1/3 π(x2)(x) = xπ/3.
Surface Area of a Cone
The surface area of a cone consists of the lateral surface area and the area of the base. The base is a circle and therefore has an area of πr2. The lateral surface is the cone “unrolled.” Depending on the shape of the cone, it can be the shape of a triangle with a curved base, a half circle, or a “Pacman” shape. The area of the lateral surface is a length that is related to the circumference of the circle times the lateral height, l. This is the formula:
where r is the radius and l is the lateral height. An alternate formula for lateral surface area is:
where c is the circumference and l is still the lateral height. The two formulas for surface area are the same, since the circumference of a circle is equal to 2πr. But knowing one or the other can save time.
The total surface area of a cone is the sum of the base area and lateral surface area:
When you are finding the surface area of a cone, be careful not to just find the lateral surface area and then stop. Students often neglect to add the area of the circular base. Practice by finding the total surface area of the cone pictured below:
The total surface area is equal to the area of the base plus the lateral surface area. The area of the base = πx2. The lateral surface area = πx 2x. The total surface area therefore equals πx2 + π2x2 = 3πx2.
Pyramids
A pyramid is like a cone, except that it has a polygon for a base. Though pyramids are not tested very often on the Math IIC test, you should be able to recognize them and calculate their volume.
The shaded area in the figure above is the base, and the height is the perpendicular distance from the apex of the pyramid to its base.
Volume of a Pyramid
The formula for calculating the volume of a pyramid is:
where B is the area of the base and h is the height. Try to find the volume of the pyramid below:
The base is just a square with a side of 3, and the height is 3/2. B = 32 = 9, and the total volume of the pyramid is:
Surface Area of a Pyramid
The surface area of a pyramid is rarely tested on the Math IIC test. If you do come across a question that covers the topic, you can calculate the area of each face individually using techniques from plane geometry, since the base of the pyramid is a square and the sides are triangles.
Spheres
A sphere is the collection of points in three-dimensional space that are equidistant from a fixed point, the center of the sphere. Essentially, a sphere is a 3-D circle. The main measurement of a sphere is its radius, r, the distance from the center to any point on the sphere.
If you know the radius of a sphere, you can figure out both its volume and surface area. The equation for the volume of a sphere is:
The equation for the surface area of a sphere is:
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