Consider the three-dimensional figure given in the diagram. By dragging the point, the figure can be skewed. However, its height and the radius of the circles that make the base and the top of the cylinder remain the same. Think about the volume — space occupied — of the solid. Does the volume change when the solid is skewed?
Before moving on, the definition of a circle and some of its parts will be reviewed.
Next, the formula for calculating the circumference of a circle will be discussed.
Since the diameter is twice the radius, the circumference of a circle can also be calculated by multiplying by
Consider two circles and their respective diameters and circumferences.
Finally, the formula for calculating the area of a circle will be seen.
A circle with radius will be divided into a number of equally sized sectors. Then, the top and bottom halves of the circle will be distinguished by filling them with different colors. Because the circumference of a circle is the arc length of each semicircle is half this value,
Now, the above sectors will be unfolded. By placing the sectors of the upper hemisphere as teeth pointing downwards and the sectors of the bottom hemisphere as teeth pointing upwards, a parallelogram-like figure can be formed. Therefore, the area of the figure below should be the same as the circle's area.
It was shown that the area of a circle is the product of and the square of its radius.
Izabella loves to use geometry in her art. Her school noticed her skills and hired her to paint the school's soccer field — for a substantial payment. The diagram shows how the field should look.
The circumference is twice the product of and the radius. The area is the product of and the square of the radius.
The circumference of a circle with radius is twice the product of and Furthermore, its area is the product of and Since the radius is meters, the circumference and area can be calculated.
The circumference and area of the circle located at the center of the soccer field are about meters and square meters, respectively.
Maya bought meters of fencing with the hopes of constructing a circular dog run for her dog Opie. Because Opie is a large Saint Bernard, she wants the area of the dog run to be at least square meters.
For the following questions, approximate the answers to one decimal place. Do not include units in the answer.
This principle will be proven by using a set of identical coins. Consider a stack in which each of these coins is placed directly on top of each other. Consider also another stack where the coins lie on top of each other, but are not aligned.
The first stack can be considered as a right cylinder. Similarly, the second stack can be considered as an oblique cylinder, which is a skewed version of the first cylinder. Because the coins are identical, the cross-sectional areas of the cylinders at the same altitude are the same.
Since the coins are identical, they have the same volume. Furthermore, since the height is the same for both stacks, they both have the same number of coins. Therefore, both stacks — cylinders — have the same volume. This reasoning is strongly based on the assumption that the face of the coins have the same area.
Consider a rectangular prism and a right cylinder that have the same base area and height.
By the Cavalieri's Principle, two solids with the same height and the same cross-sectional area at every altitude have the same volume. Therefore, the volume of the cylinder is the same as the volume of the prism. Furthermore, the volume of a prism can be calculated by multiplying the area of its base by its height. By the Transitive Property of Equality, a formula for the volume of the cylinder can be written. Finally, not only is the area of the base of the prism, but also the area of the base of the cylinder. Since the base of the cylinder is a circle, its area is the product of and the square of its radius Therefore, can be substituted in the above formula. This formula applies to all cylinders because there is always a prism with the same base area and height. Also, by the Cavalieri's principle, this formula still holds true for oblique cylinders.
LaShay loves golfing. She is about to buy a new golf bag.
The golf bag is in the shape of a cylinder. Therefore, to calculate its volume, the formula for the volume of a cylinder can be used.
Besides the radius, height, and volume, another essential characteristic of a cylinder is its surface area. The surface area of a solid is the measure of the total area that the surface of the solid occupies.
The surface area of this cylinder is given by the following formula.
Since the top and the bottom are congruent circles, they have the same area. To find the area of the rectangle, its length must be found first. The length of the rectangle that forms the lateral part of the cylinder is the circumference of the base, which is
Therefore, the area of the rectangle is the product of and The surface area of the cylinder is the sum of the areas of the rectangle and the circles.
Jordan's father is giving her a new baseball bat for her birthday. To avoid damage to the bat, Papa Jordan will store the bat in a box that has the shape of a cylinder with a height of inches and a radius of inches. Since this is a present for Jordan, Papa will use wrapping paper to make it even more special.