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 figure above, these three measurements are denoted by r, h, and l, respectively. Notice that the height, radius, and lateral height of a cone form 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 that it is 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:
Surface Area of a Cone The surface area of a cone consists of the lateral surface area and the area of the base. Because the base is a circle, it has an area of πr2. The lateral surface is the cone “unrolled,” which, depending on the shape of the cone, 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 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. The total surface area 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 find only the lateral surface area and then stop. Students often forget the step of adding on 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 area of the lateral surface. 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 IC 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:英语作文
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