8.4 Scale and Similarity

**Glacial scale (department November 2025).** On a 1960 map with scale $3$ cm : $600$ m, the rectangle outlining a glacier measures 6 cm by 9 cm. On a 2010 map with scale $5$ cm : $500$ m, the rectangle outlining the same glacier measures 4 cm by 6 cm. (a) Find the real dimensions of each rectangle in metres. (b) Find the real area of each rectangle in square kilometres. (c) Find the ratio Area(R1) : Area(R2) in simplest form. (d) Find the linear scale factor between R1 and R2. (e) Use a linear extrapolation to predict whether the glacier disappears by 2035.

**Floor plan (1 : 50).** A floor plan at 1 cm : 50 cm shows a room measuring 8 cm × 6 cm. (a) Find the real dimensions. (b) Find the real floor area in m². (c) The plan also shows a circular table (radius 0.4 cm on the plan). Find its real radius and area. (d) Show that the plan-area : real-area ratio is $1 : 2500$.

**Shadow stick.** A 1.5 m vertical stick casts a 1 m shadow at the same time of day that a tree casts a 7 m shadow. (a) Why are the shadow triangles similar? (b) Find the tree height. (c) The shadow of a flagpole is measured later in the day at 5 m, when the stick now casts a 2 m shadow. Find the flagpole height.

**Similar pictures.** A photograph is 10 cm × 15 cm. A larger printed copy is similar and has area 600 cm². (a) Find the linear scale factor. (b) Find the larger picture's dimensions. (c) If the smaller picture costs 4 chf and price is proportional to area, find the cost of the larger picture.

**Scale-model car.** A scale model of a car is 1 : 24. The real car is 4.8 m long, 1.8 m wide and 1.5 m tall. (a) Find the model's dimensions in cm. (b) Find the surface-area ratio (model : real) and the model's surface area if the real is 36 m². (c) Find the volume ratio (model : real) and the model's volume if the real has interior volume 12 m³.

**Designing a school garden.** A scale drawing of a school garden uses 1 cm : 2 m. (a) On the drawing, a flowerbed is a triangle with vertices (1 cm, 1 cm), (5 cm, 1 cm), (1 cm, 4 cm). Find the real vertices in metres (relative to the same origin). (b) Find the real area of the flowerbed. (c) A path of width 0.5 m surrounds the flowerbed on its three straight sides. Find the path width on the drawing.

**Similar shapes investigation.** Two similar shapes have areas 50 cm² and 200 cm². (a) Find the linear scale factor between them. (b) The smaller has perimeter 30 cm. Find the larger perimeter. (c) The smaller has 3 cm of trim around its border. Find the trim length on the larger shape.

**Map of Geneva.** On a 1 : 50 000 map, Cornavin station is 12 cm from UN headquarters along a straight line. (a) Find the real distance in metres and kilometres. (b) On a different map at 1 : 25 000, what would the same distance measure on the map? (c) The area of Geneva inside the city limits is approximately 16 km². How much area would this occupy on the 1 : 50 000 map (in cm²)?

**Doll-house furniture.** A doll house is 1 : 12. A real chair is 90 cm tall. (a) Find the doll-house chair height in cm. (b) A real living-room rug is 2.4 m × 1.8 m. Find the dimensions of the doll-house version. (c) The real chair weighs 8 kg. Assuming the doll-house chair is made of similar material at the same density, estimate its mass.

**Two similar boxes.** Two cuboid boxes are similar. The smaller has dimensions $a \times b \times c$ and volume $V$. The larger has linear scale factor $k$ relative to the smaller. (a) Express the larger volume in terms of $V$ and $k$. (b) The smaller box has volume 250 cm³. The larger box has volume 2000 cm³. Find $k$. (c) The smaller has surface area 200 cm². Find the surface area of the larger.

**Building's shadow.** The Manhattan skyscraper "One World Trade Center" is 541 m tall. At noon, its shadow is 380 m long. A pedestrian 1.7 m tall is nearby. (a) Find the angle of elevation of the sun to 1 d.p. (b) Find the pedestrian's shadow length at the same time (assume parallel sunlight). (c) Demonstrate that the shadow triangles are similar.

**Cake tin scaling.** A round cake tin has diameter 20 cm and depth 5 cm. A similar (linearly enlarged) cake tin is made with diameter 30 cm. (a) Find the linear scale factor. (b) Find the depth of the new tin. (c) The volume of cake batter required to fill each tin is proportional to its volume. The small tin needs 1.5 litres. Find the volume for the larger tin.