Similarity bodies - practice problems
Similarity of bodies explores the relationships between three-dimensional geometric shapes that have the same form but different sizes. Two bodies are similar when all corresponding linear dimensions are proportional by the same ratio, known as the scale factor. The concept is fundamental in understanding how volume and surface area scale with size changes. Applications include architectural modeling, map scaling, and understanding how physical properties change with size. Students learn to calculate missing dimensions, surface areas, and volumes of similar solids. This topic builds on two-dimensional similarity concepts and extends them into spatial reasoning.Instructions: Solve each problem carefully and provide a detailed solution for every item.
Number of problems found: 14
- Scale model
In a model train set, 1.38 inches represents one foot in real life. The height of One World Trade Center in New York City is 1776 feet. How tall would a scale model of the building be? Should you calculate 1776 x 1.38 or 1776 ÷ 1.38? - Eiffel Tower
The Eiffel Tower in Paris is 300 meters high and made of steel. It weighs 8,000 tons. If the tower model made of the same material weighs 2.8 kg, how tall is it? - Equilateral cone
We pour so much water into a container with the shape of an equilateral cone, the base of which has a radius r = 6 cm, that one-third of the volume of the cone is filled. How high will the water reach if we turn the cone upside down? - Sun rays
If the sun's rays are at an angle of 60°, then the famous Great Pyramid of Egypt (which is now 137.3 meters high) has a 79.3 m long shadow. Calculate the current height of the neighboring Chephren pyramid, whose shadow is measured at the same time at 78.8 - Cone
The circular cone of height 14 cm and volume 4396 cm³ is at one-third of the height (measured from the bottom) cut by a plane parallel to the base. Calculate the radius and circumference of the circular cut. - Pyramid cut
We cut the regular square pyramid with a parallel plane to the two parts (see figure). The volume of the smaller pyramid is 20% of the volume of the original one. The bottom of the base of the smaller pyramid has an area of 10 cm². Find the area of the or - Similar frustums
The upper and lower radii of a frustum of a right circular cone are 8 cm and 32 cm, respectively. If the altitude of the frustum is 10 cm, how far from the bottom base must a cutting plane be made to form two similar frustums? - Cutting cone
A cone with a base radius of 10 cm and a height of 12 cm is given. At what height above the base should we divide it by a section parallel to the base so that the volumes of the two resulting bodies are the same? Express the result in cm. - Pizza
Pizza with a diameter of 40 cm weights 409 g. What diameter will a pizza weigh 764 g made from the same cloth (same thickness) and decorated? - Hexagon cut pyramid
Calculate the volume of a regular 6-sided cut pyramid if the bottom edge is 30 cm, the top edge is 12 cm, and the side edge length is 41 cm. - Right circular cone
The volume of a right circular cone is 5 liters. The cone is divided by a plane parallel to the base, one-third down from the vertex to the base. Calculate the volume of these two parts of the cone. - Two pots
Two similar pots have 16 cm and 10 cm heights if the smaller pot holds 0,75 l. Find the capacity of the larger pot - Similarity
Rectangle ABCD has dimensions of 7 cm and 8 cm. Rectangle PQRS has dimensions 7 cm and 6 cm. Determine the coefficient of the similarity k of the rectangles; if they aren't similar, enter zero as the coefficient of similarity. - Dimensions of the frame
The picture frame is made of a 6 cm wide bar. The dimensions of the image are 74 and 57 cm. Are the inner and outer edges of the frame two similar baffles?
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