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It’s your turn on lengths, areas, and volumes

It's your turn on lengths, areas, and volumes
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Hello. Welcome to the section “It’s your turn on lengths, areas, and volumes”. OK, we have the following exercise. We have a circle.
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We know that the length on the circumference of the arc corresponding to an angle equal to pi over 6 is equal to 3 meters. OK. We want to compute the perimeter and the area. OK, of course, to compute the perimeter and the area, we have first of all to compute the radius of this circumference. OK? But what do we know? We know that the arc length on the circumference corresponding to an angle of pi over 6 radians is equal to what? To the radius times pi over 6. Therefore, we know that three is equal to the radius times pi over 6. OK, then we immediately get that the radius is equal to 18 over pi. OK.
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But then, which is the perimeter of the circumference? The perimeter is equal to 2 pi times the radius. That is, 2 pi times 18 over pi, which is 2 times 18, which is 36 meters.
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And which is the area?
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It is pi times the square of the radius. Therefore, it’s pi times 18 over pi squared. OK? And 18 squared is 324. And then we have pi over pi squared. Then it remains pi on the bottom. Of course, because it is an area, meters squared– square meters. Hello. Let us consider the second exercise of the section “It’s your turn on lengths, areas, and volumes”. OK, we have a cube. And we know the area of its one face is 90 square meters. OK, and we want to compute the volume and the surface area of the cube. OK, let us start computing the surface area, because what is the surface area?
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The surface area is equal to 6 times the area of one face. Then it’s equal to 6 times 90, which is 540 square meters.
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Very easy. And what we can say about the volume of this cube? You know, if you know the length of one side of your cube, then the volume of the cube is the cube of this length. OK, but we know that the area of one face is 90 square meters. Then the length of one side– we call it A– is equal to what? To the square root of 90, which is 3 times the square root of 10 meters. And now that we know the length of one side of our cube, then immediately we can conclude that the volume is equal to what? To A cubed, OK? Which is equal to 3 times the square root of 10 cube– cube meters.
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That is 27 times 10 times the square root of 10 meters– cube meters. That is, 270 times the square root of 10 cube meters. OK, we can compute the volume also with another reasoning. You see, when you know the area of one face of your cube, then to get the volume of this cube, it is sufficient to multiply this area– 90 square meters– by the length of one side, OK? And we have computed before the length of one side. OK, this was another way to compute the volume. And of course, in following this procedure, you get exactly the same result. Hello. Let us consider now the third exercise of the section “It’s your turn on lengths, areas, and volumes”. Good.
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We have a right circular cone. And we know the radius, which is equal to 3 centimeters. We know the sides’ area, which is equal to 15 pi square centimeters. And we would like to compute the height of our circular cone. Good. Now, remember that if you have a circular cone of radius r and height h, then the sides’ area is equal to what? To pi r times the square root of r squared plus h squared. Good. Then what do we know? We know that 15 pi has to be equal to what? To pi times the radius, which is three centimeters multiplied by the square of the radius– 9– plus h squared. Good. Now, we can simplify this expression.
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You know, we can divide by pi on both sides. And we can divide by 3 on both sides. And we get 5 equal to the square root of 9 plus h squared. And now, considering the square of both sides, what do we get? 25 equal to 9 plus h squared. Therefore, we have that h squared is equal to 25 minus 9; that is, 16. And then the height is equal to 4 centimeters. OK, thank you very much for your attention. [IN ITALIAN] Goodbye to everybody

Do your best in trying to solve the following problems. In any case some of them are solved in the video and all of them are solved in the pdf file below.

Exercise 1.

Let (mathcal C) be a circle. Assume that the arclength on the circumference corresponding to an angle of (pi/6) radians at the center is equal to (3 m). Compute the perimeter and the area of (mathcal C).

Exercise 2.

Find the volume and surface area of a cube, if the area of its one face is 90 (m^2).

Exercise 3.

Find the height of a right circular cone of radius (3 cm) and side’s area (15pi cm^2).

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