Measurement: Perimeter, Area, Surface Area, Volume & Similarity

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Presentation transcript:

Measurement: Perimeter, Area, Surface Area, Volume & Similarity Unit 1: Y11 Mathematics Applications

Unit Content Perimeter & Area 1.3.2 solve practical problems requiring the calculation of perimeters and areas of circles, sectors of circles, triangles, rectangles, parallelograms and composites Surface Area & Volume 1.3.3 calculate the volumes of standard three‐dimensional objects, such as spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations, for example, the volume of water contained in a swimming pool 1.3.4 calculate the surface areas of standard three‐dimensional objects, such as spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations; for example, the surface area of a cylindrical food container Similarity 1.3.5 review the conditions for similarity of two‐dimensional figures, including similar triangles 1.3.6 use the scale factor for two similar figures to solve linear scaling problems 1.3.7 obtain measurements from scale drawings, such as maps or building plans, to solve problems 1.3.8 obtain a scale factor and use it to solve scaling problems involving the calculation of the areas of similar figures and surface areas and volumes of similar solids

Unit Conversions: Length Example 230.5m = _________________ km 0.054m = _________________ mm

Perimeter and Circumference Perimeter is the measure of the distance around the outside of a two dimensional shape. The circumference is the perimeter of a circle. Circumference = 2 × π × r Circumference = π × D

Perimeter: Sectors Perimeter of Sector =

Perimeter: Sectors Example

Perimeter Example A sports ground 140m long and 50m wide has semicircular ends. A fence is to be built all the way around, with four gates each 2m wide. The gates cost $150 each and the fencing costs $45/m. Calculate the cost of fencing the ground, correct to the nearest dollar.

Metric Conversions: Area Example 55m2 = ______________ cm2 548ha = ______________ km2

Area

Area Example

Area: Sectors Area of a Sector = Example

Area: Composite Shapes Example

Area: Composite Shapes Example

Surface Area The surface area of a solid is the total area of all of its faces.

Surface Area: Prisms Example

Surface Area: Cylinders Open at both ends: Open at one end: Closed:

Surface Area: Cylinders Example Calculate the surface area of this open-top cylinder.

Surface Area: Pyramids Example

Surface Area: Cones

Surface Area: Spheres Example

Surface Area: Composite Solids Example

Surface Area: Composite Solids Example

Surface Area: Composite Solids Example

Metric Conversions: Volume & Capacity Example 74m3 = __________________ cm3 4600mm3 = ____________________cm3 3500m3 = _____________________ ML

Volume and Capacity Volume is the amount of space that a solid occupies, it is measured in cubic units. The capacity of a solid is a measure of how much liquid it can contain, it is usually measured in millilitres and litres.

Volume

Volume: Prisms Example

Volume: Cylinders Example

Volume: Pyramids Example

Volume: Cones Example An ice-cream cone has a diameter of 7cm and a slant height of 17cm. What is the perpendicular height of the cone? How many millilitres of ice-cream can the cone hold if it is only filled level with the top of the cone?

Volume: Spheres Example

Volume: Composite Solids Example

Volume: Composite Solids Example

Scales Scale = drawing length : real length

Scales Example The scale on a map was 1: 100 000. What was the real distance between two shops which were 8cm apart on the map? How long was the line on the map indicating a 6.8km straight stretch of road?

Scale Drawings Example How long is the microorganism in the drawing? How long is the microorganism in real life?

Scale Drawings Example

Scale Drawings Example What is the length of the verandah in metres? What is the cost of tiling the family room if it costs $62.50 per m2 laid?

Scale Drawings: Choosing a Scale Example A house that is 16m long and 9m wide is to be drawn on A4 paper. What scale should be used?

Similar Figures Similar figures have the same shape but not necessarily the same size. They have: Corresponding angles equal Corresponding sides in the same ratio

Similar Figures: Scale Factor Scale Factor = length in image ÷ corresponding length in object Example Calculate the scale factor Calculate y Calculate x

Similar Triangles: Tests for Similarity SSS – all ratios of corresponding side lengths are equal SAS – two pairs of corresponding sides in the triangles are in the same ratio and the angles between them are equal AAA – the angles of each triangle are the same RHS – the hypotenuses and a pair of corresponding sides are in the same ratio in two right-angled triangles

Similar Triangles: Tests for Similarity Example

Similar Triangles Example Give a reason why these two triangles are similar. Calculate the value of x.

Similar Triangles: Applications Example A stick of height 1.5m casts a shadow of length 3.8m. At the same time, a tree casts a shadow 31m long. What is the height of the tree?

Similar Triangles: Applications Example A surveyor needs to measure the distance across a river. There are two trees on the opposite bank that are 45m apart. She stands 5m from the bank, directly opposite the first tree. Her assistant has to move 8.4m long the bank to place a stick directly in her line of sight to the second tree. Find the width of the river.

Areas of Similar Figures Length Scale Factor = Length Scale Factor = If the length scale factor is k, then the area scale factor is _______

Areas of Similar Figures Example Two regular pentagons have side lengths of 4cm and 2.5cm respectively. Length scale factor = Area scale factor =

Areas of Similar Figures Example A photograph is 8cm long. Its enlargement is 12cm long. If the area of the small photograph is 48cm2, what is the area of the enlargement?

Surface Areas of Similar Solids Example What is the length scale factor? What is the area scale factor? The surface area of the larger prism is 2400cm2, what is the surface area of the smaller prism?

Volumes of Similar Solids In general, if the length scale factor is k, then the volume scale is __________

Volumes of Similar Solids Example A square based pyramid with a base of 4cm is enlarged to produce a similar pyramid with a base length of 10cm. What is the length scale factor? What is the volume scale factor? If the volume of the smaller pyramid is 32cm3, what is the volume of the larger pyramid?

Volumes of Similar Solids Example A chocolate bar is is the shape of a rectangular prism. The manufacturer wants to increase profits by reducing the size of the bar. If the dimensions of the bar are reduced by one quarter, what is the ratio of the volume of the new bar compared to the volume of the original? If the original bar wads 320g, what is the width of the new bar?