Surface Area and Volume

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Questions and Answers

A rectangular prism has dimensions 5 cm x 4 cm x 3 cm. If each dimension is doubled, by what factor does the surface area increase?

The surface area increases by a factor of 4.

A sphere has a volume of $36\pi$ cubic inches. What is its surface area, in square inches?

$36\pi$ square inches.

A right circular cylinder has a radius of 2 cm and a height of 5 cm. A cone has the same radius, but its volume is equal to that of the cylinder. What is the height of the cone?

15 cm

A cube has a surface area of 150 square cm. What is the volume of the cube?

<p>125 cubic cm</p> Signup and view all the answers

A cone has a radius of 3 inches and a slant height of 5 inches. What is the lateral surface area of the cone?

<p>$15\pi$ square inches.</p> Signup and view all the answers

A solid is formed by a cylinder of radius $r$ and height $2r$ capped by a hemisphere of radius $r$. What is the total volume of the solid?

<p>$3\pi r^3$</p> Signup and view all the answers

Two spheres have radii in the ratio of 1:2. What is the ratio of their surface areas?

<p>1:4</p> Signup and view all the answers

A swimming pool is a rectangular prism with dimensions 10m x 5m x 2m. How many liters of water are needed to fill it completely? (Note: 1 cubic meter = 1000 liters)

<p>100,000 liters</p> Signup and view all the answers

A waffle cone has a diameter of 8 cm and a height of 15 cm. If the cone is filled with ice cream, what is the volume of the ice cream?

<p>$80\pi \text{ cm}^3$</p> Signup and view all the answers

A regular hexagonal prism has a base area of $24\sqrt{3}$ square inches and a height of 10 inches. What is the volume of the prism?

<p>$240\sqrt{3}$ cubic inches.</p> Signup and view all the answers

Flashcards

Surface Area

The total area of all the surfaces of a three-dimensional object.

Volume

The amount of space a three-dimensional object occupies.

Cube Surface Area

6 times the side length squared: SA = 6s^2

Rectangular Prism Surface Area

2(lengthwidth + lengthheight + width*height): SA = 2(lw + lh + wh)

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Sphere Surface Area

4 * pi * (radius)^2: SA = 4Ï€r^2

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Cone Surface Area

pi * (radius)^2 + pi * radius * slant height: SA = πr^2 + πrl

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Cube Volume

(side length)^3: V = s^3

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Rectangular Prism Volume

length * width * height: V = lwh

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Sphere Volume

(4/3) * pi * (radius)^3: V = (4/3)Ï€r^3

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Cone Volume

(1/3) * pi * (radius)^2 * height: V = (1/3)Ï€r^2h

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Study Notes

  • Surface area represents the total area of all the surfaces of a 3D object
  • Volume measures the amount of space that a 3D object occupies

Surface Area of a Cube

  • A cube consists of 6 faces, all of which are squares
  • Surface Area = 6 * (side length)^2
  • SA = 6s^2

Surface Area of a Rectangular Prism

  • Rectangular prisms have 6 faces, and all of them are rectangles
  • Surface Area = 2 * (length * width + length * height + width * height)
  • SA = 2(lw + lh + wh)

Surface Area of a Sphere

  • Surface Area = 4 * pi * (radius)^2
  • SA = 4Ï€r^2

Surface Area of a Cylinder

  • A cylinder includes two circular faces and a curved surface
  • Surface Area = 2 * pi * (radius)^2 + 2 * pi * radius * height
  • SA = 2Ï€r^2 + 2Ï€rh

Surface Area of a Cone

  • A cone has a circular base and a curved surface
  • Surface Area = pi * (radius)^2 + pi * radius * slant height
  • SA = Ï€r^2 + Ï€rl

Volume of a Cube

  • Volume = (side length)^3
  • V = s^3

Volume of a Rectangular Prism

  • Volume = length * width * height
  • V = lwh

Volume of a Sphere

  • Volume = (4/3) * pi * (radius)^3
  • V = (4/3)Ï€r^3

Volume of a Cylinder

  • Volume = pi * (radius)^2 * height
  • V = Ï€r^2h

Volume of a Cone

  • Volume = (1/3) * pi * (radius)^2 * height
  • V = (1/3)Ï€r^2h

Units of Measurement

  • Area is measured in square units (e.g., cm^2, m^2, in^2)
  • Volume is measured in cubic units (e.g., cm^3, m^3, in^3)

Lateral Surface Area

  • Lateral surface area refers to the area of the sides of a 3D object, excluding the base(s)
  • For a cylinder, the lateral surface area is 2 * pi * radius * height or 2Ï€rh
  • For a cone, the lateral surface area is pi * radius * slant height or Ï€rl

Composite Shapes

  • Composite shapes are made up of two or more basic shapes
  • To find the surface area of a composite shape, calculate the surface area of each individual shape and then add them together, subtracting any overlapping areas
  • To find the volume of a composite shape, calculate the volume of each individual shape and then add them together

Cavalieri's Principle

  • If two solids have the same height and the same cross-sectional area at every level, then they have the same volume
  • This principle is useful for finding the volume of oblique shapes

Scaling

  • If you scale a 3D object by a factor of k, the surface area is scaled by a factor of k^2, and the volume is scaled by a factor of k^3

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