In this lesson we will be answering questions about circles and solids.
Let's first review the definition of a circle. A circle is a plane figure that
consists of all points that are the same fixed distance from a point c. The
point C is called the center.
So this blue figure here is a circle, this point is the center. The radius of
a circle is the distance from the center of the circle to any point on the circle,
which would be this distance here. The diameter of a circle is the distance
across the circle passing through the center, which would be this distance here.
The diameter is twice the radius and the radius is half the diameter. So if we let
r equal the radius
and d equal the diameter,
because the diameter is twice
the radius we can write the equation d equals two times r, or d equals two-r.
And because the radius is half the diameter,
we can write the equation r equals d divided by two.
These equations might be helpful when answering our questions about a circle.
Going back to the previous slide. In the first circle we're asked to find the diameter.
Looking at the given information notice how we're given the radius is equal to four
centimeters. And we're looking for the length of the diameter, which would be the
length all the way across the circle passing through the center.
By looking at the figure we should be able to recognize the diameter is equal
to two times four centimeters, or eight centimeters. But if we want to write an
equation, we could write the diameter d is equal to two times r, where r is equal to
four centimeters. Two times four centimeters is equal to eight centimeters, which is the diameter.
The next circle, we're given the length of segment
RJ is ten inches, which means the diameter is ten inches and we're asked to
determine the radius. So looking at the figure notice how we can see the radius
is equal to half the diameter and therefore,
the radius is equal to ten inches divided by two, or five inches. But if we want to show
some work, we can say the radius is equal to d divided by two, which is ten inches,
divided by two and therefore, r is equal to five inches.
Now the third question deals with a sphere. The relationship between the
radius and diameter of a sphere is the same relationship as the radius and
diameter for a circle.
And here we're given the diameter of a sphere is twenty-five miles,
find the radius.
So here we'll use the equation for the radius. The radius is equal to the diameter
divided by two. In this case the radius is equal to twenty-five miles
divided by two. Let's find this quotient here on the side.
Twenty-five divided by two, that's one, two and two. One times two is two, we subtract. The difference is zero.
Bring down the five.
There are two twos in five, two times two is four, we subtract. Subtract, the difference
is one.
So the quotient is twelve and one-half, which means the radius is twelve and
one-half, but let's also express this using a decimal.
So I'll put a decimal to the right of the five and then up into the quotient.
And now we can add a zero and bring it down. And now determine how many twos in ten which
is five. Five times two is ten, subtract, the difference is zero. The quotient is twelve point five and
therefore, the radius is twelve point five miles.
And now for the next question we are asked to identify each solid. Referring
to our notes,
here we have a rectangular solid, a cube,
a square pyramid, a sphere,
a cylinder and a cone.
So you may want to pause the video and read some of these definitions.
Going back to our question. Looking at our first figure here we have two
parallel bases that are circles and therefore, this is a cylinder. A cylinder is a
solid with curved sides and two parallel bases that are circles.
Looking at the next solid we need to decide whether this is a rectangular solid or a
cube. A rectangular solid is a solid that consists of six sides, or faces, all of
which are rectangles. And cube is a rectangular solid whose six sides are
squares.
We're not given any measurements of these sides and therefore, we cannot
assume all of the faces are squares, so we'll identify this is a rectangular solid.
Now looking at figure c,
notice how this solid does resemble a square pyramid where a square pyramid is
a solid with a square base and triangular sides, which meet at a point
called the apex.
But our base is not a square our bases a triangle and therefore, the name of this
solid is a triangular pyramid.
I hope you found this helpful.