7.07 Similar areas and volumes

Find the area scale factor for each of the following linear scale factors:

Consider the following two triangles:

Find the value of x for each of the following pairs of similar shapes:

a

b

A square with area 4 \text{ cm}^2 has its side lengths enlarged by a factor of 3.

The area of a square is 30 \text { cm}^2. Its side length is enlarged by a scale factor of 3. What is the area of the new square?

A square of side length 6 cm is enlarged using the scale factor 3. Find the area of the enlarged square.

The corresponding sides of two similar triangles are 8 cm and 40 cm. If the area of the smaller triangle is 24 \text{ cm}^2. Find the area of the larger triangle.

If the diameter of a circle is tripled, what will happen to the area of the circle?

A triangle has side lengths of 7\text{ cm}, 10\text{ cm} and 16\text{ cm}. A similar triangle has 9 times the area of the first triangle.

Consider two similar parallelograms with a matching sides in the ratio 6:8.

Consider the formula A = \dfrac{1}{2} b h.

Consider the formula A = \pi r^{2}.

Two similar cones have bases with radius 7\text{ cm} and 28\text{ cm} respectively.

The bases of two similar rectangular prisms are: 30\text{ cm} by 45\text{ cm}; and 6\text{ cm} by 9\text{ cm}.

The surface areas of two similar triangular prisms are in the ratio 64:49.

The volume of two similar crates are in the ratio 1331:125.

A circular oil spill has a radius of 20 \text{ m}. In a photo taken of the oil spill, the circle is reduced by a factor of 500. Find the radius of the circular oil spill in the photo.

William recorded a video on his camera. When viewing it on the camera screen, the video appeared in a width to height ratio of 4:3 respectively. When he uploaded to his computer, everything stayed in the same ratio. If the width of the video on his computer screen is 20 \text{ cm}, find the height of the video on the computer screen.

A rectangular billboard has a length of 1.2\text{ m}. The corresponding length on the billboard designer's computer screen is 20\text{ cm}.

A piece of sports tape in the shape of a rectangle measures 6\text{ cm} in width and 10\text{ cm} in length (when not stretched). When applied to Sophia’s shoulder, it is stretched so that it covers a rectangular area measuring 12\text{ cm} wide by 20\text{ cm} long.

Mae wants to insert a picture into a document. She enlarges it by a factor of 12 but it becomes too blurry, so she reduces the resulting picture by a scale factor of 4. What is the overall scale factor going from the original to the final size?

Valentina was making a trial birthday cake for her son. The dimensions are 10\text{ cm} for the length, 7\text{ cm} for the width and 3\text{ cm} for the height. The final cake needs to have dimensions of 20\text{ cm} for the length, 14\text{ cm} for the width and 6\text{ cm} for the height.

Susana has two teddy bears that have the same shape but are different sizes. The length of the first teddy bear is 15\text{ cm}, while the length of the second teddy bear is 75\text{ cm}.

The dimensions of a cement slab are l, the length, w, the width, and h, the height. If these dimensions are tripled:

The radii of two spherical balloons are 12\text{cm} and 6\text{cm} respectively.

Consider two similar ceilings: the first with dimensions 5\text{ m} by 4\text{ m}, and the second with dimensions 20\text{ m} by 16\text{ m} .

The ratio of the length of a model car to a real car is 1:20.

A beehive consists of hexagonal cells with side length 3.2\text{ mm} and depth of 3.8\text{ mm}. A plastic container is built which is modelled on these hexagonal cells, with a side length of 6.4\text{ cm}.