Volume Equation of Geometrical Figures
Before learning the volume equation of various geometrical figures, let’s list these figures and then study their formulas. Each geometrical body has a surface, and if it consists of flat polygons, such a body is called a polyhedron, and polygons constituting its surface are faces. The boundaries between the facets are called edges, and the points, at which the ribs are connected, are vertices of polyhedron.
Thus, polyhedrons are the bodied bounded by plane polygons. They are everywhere around us: in fact the most popular form of modern building, TV, or furniture is a parallelepiped.
Calculation of the Volume of Geometrical Bodies Using Integrals
- Cube volume equation
Cube is a three-dimensional shape that has equal height, width, and measurements of length. A cube has six faces that are square with all sides of equal length and meet at right angles. To find a volume of a cube you need to multiply its length by height and by width: V = S3
- Now let’s see how to calculate the volume of a cuboid.
Choose a rectangular system of coordinates in the space Oxyz, where A is a permissible cuboid (a cuboid, which sides are parallel to coordinate axes), and the lengths of the edges are a, b, c. Let’s name the number abc the volume of the cuboid and let’s mark it with V (A) = abc. Obviously, if cuboid A is divided by a plane parallel to one of the coordinate planes, into cuboids B and C, then the following equality is true: V (A) = V (B) + V (C).
If cuboid A’ is obtained from the cuboid A by parallel transfer, then V (A ‘) = V (A). Finally, the volume equation of a cube with the length of the edge 1 is equal to 1.
- Volume Equation of Right Cylindrical Body
Let’s assume that F is a plane figure. Let’s draw a perpendicular at every point of this figure to its plane and mark off on every segment of length the perpendicular h (all segments are arranged on one side of the plane). The plurality of the points of these segments forms a body L, which is called a direct cylindrical body with a base F and the height h. The second ends of the constructed segments form a figure F *, which is congruent to the base F and parallel to it.
When F is a rectangle, right cylindrical body is a rectangular parallelepiped. If F is a stepped figure, then L is the stepped body, wherein it is decomposed into rectangular parallelepipeds having the same heights. The volume of this stepped body is the product of the square area of F by the body height: V (L) = S (F)h.
- Volume Equation of Cylinder
Cylinder is a volume geometric body with parallel bases of the round form. If you need to calculate the volume of a cylinder, find its height (h) and the base radius (r), and then substitute them into the volume equation: V = hπr2.
At first, find the base radius. Both bases of a cylinder are equal. If the radius is given, go to the next step. Otherwise, measure the circle in its widest part, to find the diameter. Then divide the diameter by 2 to acquire the radius. For example, the radius of the cylinder is 1 cm.
- If the value of diameter if given, divide it by 2 to get the radius.
- If the circumference is given, divide it by 2π to get the radius.
Find the square of the base according to the formula: A = πr2. Just put the value of the radius in the formula. Here’s how: A = π x 12 =A = π x 1. Since π ≈ 3,14, the base area is 3.14 cm2.
The next step is to find the height of the cylinder. If it is given, go to the next step. Otherwise measure it with a ruler. Height is the distance between the two bases. For example, the height is 4 cm. Multiply the area of the base by the height of the cylinder to find its volume. The area of the base is equal to 3.14 cm 2, and the height is 4 cm, so the 3.14 cm2 x 4 cm = 12.56 cm3. The volume is measured in cubic units, as this quantity characterizes volumetric (three dimensional) shapes.
- Volume Equation of the Right Pyramid
Pyramid is a three-dimensional figure whose base is a polygon, and the faces are triangles that have a common vertex. The right pyramid is a three-dimensional figure whose base is a right polygon (with equal sides), and the top is projected in the center of the base.
Typically we imagine a pyramid having a square base, but the base of the pyramid can have a polygon with 5, 6 or even a 100 sides. Pyramid with round base is called a cone.
The volume equation for the right pyramid is: V = 1 / 3BH, where B is the area of the base of the pyramid, and h is the height of the pyramid (the perpendicular connecting the base and the top of the pyramid).
The volume equation for calculating the volume of the pyramid is suitable for both right pyramids (when the vertex is projected in the center of the base) and for inclined pyramids (when the vertex is not projected in the center of the base).
To find the volume of the right pyramid, first calculate the area of the base. The volume equation will depend on the figure lying in the base of the pyramid. Let’s assume that there is a square with sides of 6 cm at the base of the pyramid. The area of a square is equal to S2, where S is the side of the square. Thus, in our example, the base area of the pyramid is 62 = 36 cm2.
If there is a tringle in the base of the pyramid, then the area of a triangle is equal to 1 / 2BH, where h is the height of the triangle, and b is the side with this height.
If any right polygon is in the base of a pyramid, then the area of any right polygon can be calculated by the formula: A = 1 / 2pa, where A is the area, p is the perimeter, and a is the apothem (a segment connecting the center of the figure to the middle of each side of the figure).
Now, find the height of the pyramid. Let’s assume that the height is given and equals to 10 cm. Multiply the area of the base of the pyramid by its height, and then divide the result by 3 to find the volume of the pyramid. The formula for calculating the volume of a pyramid is: V = 1 / 3bh. In our example, the base area is 36, and the height is 10, so the volume: 36 * 10 * 1/3 = 120.
If, for example, there is a pentagonal pyramid with the base area 26, and the height of the pyramid is equal to 8, the volume of the pyramid: 1/3 * 26 * 8 = 69.33.
- Volume Equation of the Cone
A cone is a three-dimensional figure, which has a circular base and one vertex, also cone is a special case of a pyramid with a circular base.
If the top of the cone is directly over the center of the circular base, the cone is called direct; otherwise it is an oblique cone. But the volume equation for the cone is the same for both types of cones.
The volume equation for the cone is: V = 1 / 3πr2h, where r is the radius of the circular base, h is the height of the cone.
The formula b = πr2 is the area of a circular base of the cone. Thus, the volume equation for calculating the volume of a cone can be written as: V = 1 / 3BH, which coincides with the formula of finding the volume of a pyramid.
First, calculate the area of the circular base. The radius should be given in the problem. If the diameter is given, then remember that d = 2r. You need to divide the diameter in half to find the radius. To calculate the area of a circular base, put the value of the radius in the formula pr2.
For example, the radius of the circular base of the cone is equal to 3 cm. Then the area of the base is π32.
π32 = π (3 * 3) = 9π. = 28.27 cm2
Then find the height of the cone. This is the perpendicular dropped from the top to the bottom of the pyramid. In this example, the cone height is 5 cm.
Multiply the height of the cone and the base area. In our example, the base area is 28.27 cm2 and the height is 5 cm, so bh = 28,27 * 5 = 141.35.
Now multiply the result by 1/3 (or simply divide it by 3), to find the volume of a cone. The volume of a cone is always 3 times less than the volume of a cylinder.
In our example: 141.35 * 1/3 = 47.12 – is the volume of the cone. Or: 1 / 3π325 = 47.12.
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