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14:00 - 15:30 Volumes and Surface Areas

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14:00 - 15:30

Volumes and Surface Areas

Surface areas and volumes

- Equal surface areas implies equal volume
Cognitive Conflict needed!

In pairs

- Use a sheet of A4 paper to make a triangular prism. Write your first names in large letters on two of the rectangular faces and place it on the table in front of you.
Manipulating - Getting a sense of – Articulating

- Which way did you fold the paper?
- What difference did it make to the area for your names?

We will be looking at the following 3D shapes:

- Prisms
- Pyramids
- Spheres

Cylinder

Cuboid

Triangular Prism

Trapezoid Prism

Surface Area of Prism = Total of ALL individual surfaces

- Can you derive a formula to calculate the surface area of a cone?

Area of circle of radius l

Fraction of circle of radius l

l

Slope height

r

2r

Surface Area of a Cone

CONE

l

SECTOR

Curved Surface area of cone = area of sector

2r

2r

2r

The Surface Area of a Sphere

The formula for the surface area of a sphere was discovered by Archimedes. In the diagram below a cylinder just encloses a sphere of radius r. Archimedes was able to determine the formula by showing that a pair of parallel planes perpendicular to the vertical axis of the cylinder, would enclose equal areas on both shapes.

Surface area = 4r2

Surface area = 2r x 2r

r

We will be looking at the following 3D shapes:

- Prisms
- Pyramids
- Spheres

Cylinder

Cuboid

Cross section

Triangular Prism

Trapezoid Prism

Volume of Prism = length xCross-sectional area

Volume of sphere = r3

= 1½ x r3

In determining the formulae for the surface area and volume of a sphere, Archimedes discovered the extraordinary fact that if you envelop the sphere perfectly with a cylinder, the cylinder will have both a volume and a surface area that are exactly 1½ times those of the sphere. He was so overjoyed at discovering this remarkable relationship between these shapes that he had them inscribed on his tombstone together with the ratio 3:2.

1. Using the appropriate formulas, establish the truth of this relationship for volume.

Volume of cylinder = r2 x 2r

r

= 2r3

SA = 4r2

The Volume of a Sphere

The formulas for the surface area of a sphere and volume of a pyramid can be used to help derive the formula for the volume of a sphere. Imagine the sphere to be composed of square - based pyramids with their bases laying on the surface and their vertices meeting at the centre. By allowing the base areas of the pyramids to become infinitely small and the number of pyramids to become infinitely large, the total base area of all the pyramids tends to 4r2. The height of each pyramid will get closer to the radius of the sphere. Therefore the total volume of all the pyramids approaches ever more closely, the volume of the sphere.

Filling a sphere with increasing numbers of smaller and smaller pyramids.

- Draw a set of axes xOy
- In the first quadrant, draw a right angled triangle, with the hypotenuse sloping forward.
- Now rotate this triangle about the x-axis through an angle of 360degrees
- What shape do you get?
- What 2D shapes should you rotate to get a cylinder? How about a frustum of a cone?

In pairs

- Use a sheet of A4 paper to make a triangular prism. Write your first names in large letters on two of the rectangular faces and place it on the table in front of you.
Manipulating - Getting a sense of – Articulating

- Which way did you fold the paper?
- What difference did it make to the volume of the prism?

Regular Polyhedra or Platonic Solids

- Tetrahedron (3 equilateral triangles at each vertex) – 4 faces
- Octahedron (4 equilateral triangles at each vertex) – 8 faces
- Icosahedron (5 equilateral triangles at each vertex) -20 faces
- Cube (3 squares at each vertex) – 6 faces
- Dodecahedron (3 regular pentagons) – 12 faces
- Paper folding on Nrich

For all polyhedra,

V+F=E+2