Total Area & Volume.

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Chapter 12 – Surface Area and Volume of Solids

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Presentation transcript:

Total Area & Volume

Rectangular Prism – A three dimensional object with two rectangular bases and four rectangular lateral faces. Base (B) Base (B)

Total Area – the sum of the areas of all of the faces (bases and lateral faces) of a solid. Lateral Area – the sum of the areas of the lateral faces of a prism

Formula for Lateral Area: Formula for Total Area: Lateral Area = Perimeter of Base x Height L.A. = ph Total Area = Lateral Area + 2(Area of the Base) T.A. = L.A. + 2B

Volume - the amount of space that an object occupies The units for Volume are always cubed. Examples: in3, m3, cm3. Formula for Volume of a Right Prism: V = Area of the base x height V = Bh

We use the same formulas for lateral area, surface area and volume when dealing with other right prisms. L.A. = ph T.A. = L.A. + 2B V = Bh Triangular Prism Trapezoidal Prism Hexagonal Prism

Pyramids – Day 2 A regular square pyramid has a square base. Slant Height (l) Height (h) Base Edge (e)

Pyramid Formula for Volume Volume = Area of the Base x Height Formula for Lateral Area L.A. = ½ Perimeter of Base x Slant Height Pyramid Formula for Total Area Total Area = Lateral Area + Area of Base

Therefore, to calculate Total Area and Volume of a Pyramid you must find four key pieces of information: 1. Area of the Base – e2 2. Perimeter of the Base – 4e 3. Height of the object – h 4. Slant Height - l

Cylinders – Day 3 Cylinders – Cylinders are very similar to the right prisms that we have been examining. The only difference is that instead of polygons (rectangle, triangle, trapezoid, hexagon) as bases, a cylinder has circular bases. The formulas to calculate lateral area, turface area, and volume will be nearly the same as prisms.

The formula for Volume remains the same. (V = Bh) The formula for Volume remains the same. (V = Bh). Because in this case the base is a circle, we must use the formula for finding area of a circle. Recall that area of a circle is calculated by using A = pr2 The Lateral Area and Total Area are calculated in a similar manner. However we must replace “perimeter of base” with circumference of base.

Therefore, to calculate Total Area and Volume of a cylinder you must find three key pieces of information: 1. Area of the Base – pr2 2. Circumference of the Base – 2pr 3. Height of the object - given

Slant Height (l) Height (h) Radius (r) A cone has one circular base. Cones – Day 3 A cone has one circular base. Slant Height (l) Height (h) Radius (r)

Cone Formula for Volume Volume = Area of the Base x Height Formula for Lateral Area L.A. = ½Circumference of Base x Slant Height Cone Formula for Total Area (Surface Area) Total Area = Lateral Area + Area of Base

Therefore, to calculate Total Area and Volume of a Cone you must find four key pieces of information: 1. Area of the Base – pr2 2. Circumference of the Base – 2pr 3. Height of the object – h 4. Slant Height - l

Spheres

Sphere – the set of all points a given distance away from a center point. Volume - Total Area -

Similar Solids Theorem 12-1 If the scale factor of two similar solids is a:b, then 1. The ratio of their perimeters is a:b. 2. The ratio of their base areas, lateral areas, and total areas is a2:b2. 3. The ratio of their volumes is a3:b3.

The diameter of a spherical water tank is 25 ft The diameter of a spherical water tank is 25 ft. There are approximately 7.48 gallons of water in 1 cu ft. How many gallons of water does the tank hold?

A teepee in the shape of a cone, has a diameter of 10 ft and a slant height of 15 ft. How much canvas is needed to cover the teepee?

The great pyramid at Giza, Egypt was built as a square pyramid with each side of its base about 756 ft. The original slant height of the pyramid was 612 ft. How many square ft of bricks were used in the construction?

A storage tank in the shape of a cylinder has a diameter of 24 ft A storage tank in the shape of a cylinder has a diameter of 24 ft. The height of the tank is 16 ft. A circular walkway 2 ft wide surrounds the tank. A) How many ft of railing are needed for the walkway around the tank? B) How many gallons of water can the tank hold? (1 gal = 0.1337 cu ft.)