When we think about mountains, we often imagine them as being narrower at the top and wider at the bottom. This shape helps them stay stable and not collapse. However, this doesn’t mean that mountains always have less land at the top. For most land creatures, the important thing is the surface area, not the volume. Volume matters more if you’re a mining company planning to dig up the entire mountain. But for the rest of us, surface area is what counts.
Interestingly, the amount of land on a mountain doesn’t always decrease as you go higher. This is especially true when the mountain is part of a mountain range. Simple, lone mountains that look like cones or spikes do have less surface area higher up. However, a mountain shaped like a parabola has more area higher up compared to a spikey mountain. Broader, flatter mountains can actually have more surface area as you climb, at least until you reach the very top. These mountains get skinnier as they rise, but they flatten out so quickly that they end up having more surface area at the top than at the bottom!
When mountains are grouped together into ranges, things get even more complicated. Some mountain ranges have less land area as you go higher, while others have more. Some ranges have more area at both the bottom and the top, with less in the middle. If you look at mountain ranges around the world, you’ll find that only about a third of them consistently have less land as you go higher. The rest have one of the other unusual “top-heavy” patterns.
So, despite what we might think, most mountain ranges are actually bigger near their tops. This has interesting implications for animals and people who might want to move up or down the mountains, especially if the climate changes.
Here’s a cool fact: a perfectly hemispherical mountain, although impossible in real life, would have the same amount of surface area at every height. This is because it gets skinnier at the same rate that it flattens out. The same math explains why if you slice an orange evenly, each piece will have roughly the same amount of skin, but different amounts of fruit inside.
Create a model of different mountain shapes using clay or playdough. Try to make a cone-shaped mountain, a parabolic mountain, and a flat-topped mountain. Observe and compare the surface areas at different heights. Discuss with your classmates which shape seems to have more surface area at the top and why.
Using graph paper, draw cross-sections of different mountain shapes at various heights. Calculate the surface area for each section and plot your findings on a graph. Analyze the graph to see how the surface area changes as you move up the mountain.
Research a specific mountain range and create a presentation about its shape and surface area distribution. Include information on how the shape affects the local climate, wildlife, and human activities. Present your findings to the class.
Simulate the impact of climate change on mountain habitats using an online tool or app. Observe how changes in temperature and precipitation affect the surface area available for plants and animals at different elevations. Discuss your observations with the class.
Write a short story from the perspective of an animal living on a mountain with an unusual shape. Describe how the shape of the mountain affects its daily life, including finding food, shelter, and avoiding predators. Share your story with the class.
Surface Area – The total area of the exterior surfaces of a three-dimensional object. – To find the surface area of a cube, you need to calculate the area of all six faces and add them together.
Volume – The amount of space occupied by a three-dimensional object, measured in cubic units. – The volume of a cylinder can be found using the formula V = πr²h, where r is the radius and h is the height.
Mountain – A large natural elevation of the earth’s surface rising abruptly from the surrounding level; often used as a metaphor for a steep slope in graphs or functions. – In the graph of a quadratic function, the vertex can be thought of as the mountain peak if the parabola opens downwards.
Range – The set of all possible output values of a function, or the difference between the highest and lowest values in a data set. – The range of the function f(x) = x² is all non-negative real numbers.
Parabola – A symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side; the graph of a quadratic function. – The path of a projectile under the influence of gravity is a parabola.
Shapes – Forms or outlines of objects, often studied in geometry, such as circles, triangles, and squares. – In geometry class, we learned how to calculate the area of different shapes like rectangles and triangles.
Height – The measurement from base to top or (in a three-dimensional object) from base to apex. – The height of a triangle is necessary to calculate its area using the formula 1/2 * base * height.
Land – The part of the earth’s surface that is not covered by water, often used in math problems to describe areas or plots. – The farmer calculated the area of his land to determine how much seed he needed to plant.
Creatures – Living beings, often used in math problems as hypothetical subjects to illustrate concepts like population growth or distribution. – In a math problem, we calculated how many creatures could live in a habitat given its size and resources.
Climate – The weather conditions prevailing in an area in general or over a long period, sometimes used in mathematical models to predict changes. – Scientists use mathematical models to predict how climate change might affect sea levels.
