Question

A regular pentagon and a regular decagon have the same perimeter. Then the ratio of their areas is (a) \(\sqrt{5}: 2\) (b) \(3: \sqrt{5}\) (c) \(5: 2\) (d) \(2: \sqrt{5}\)

Short answer

Answer: The ratio of the areas is (d) \(2: \sqrt{5}\).

Step-by-step solution

Write down the perimeter and area formula for a regular pentagon and a regular decagon

For both regular pentagon and regular decagon, the perimeter is the sum of all the side lengths. Let the side length of the regular pentagon be \(a\) and that of the regular decagon be \(b\). The perimeter of the pentagon \(P = 5a\) and the perimeter of the decagon \(Q = 10b\). The area of a regular pentagon is given by \(A_{pentagon} = \frac {a^2 \sqrt{25 - 10\sqrt{5}}}{4}\), and the area of a regular decagon is given by \(A_{decagon} = \frac{b^2 \sqrt{500 - 50\sqrt{5}}}{4}\).

Establish the relationship between the side lengths of the pentagon and decagon

Since the perimeters of the regular pentagon and regular decagon are equal, we have \(P = Q \Rightarrow 5a = 10b\). Dividing both sides by 5, we get \(a = 2b\).

Express the area of pentagon in terms of the area of decagon

Substitute the value of \(a = 2b\) in the area formulas of the regular pentagon and regular decagon. Now, we can write the area of the regular pentagon as a function of the area of the regular decagon: \(A_{pentagon} = \frac {[2b]^2 \sqrt{25 - 10\sqrt{5}}}{4}\). Now we can find the ratio between the areas. Ratio of areas: \(\frac{A_{pentagon}}{A_{decagon}} = \frac{\frac{[2b]^2 \sqrt{25-10\sqrt{5}}}{4}}{\frac{b^2 \sqrt{500 - 50\sqrt{5}}}{4}}\). The \(b^2\) and the factor \(\frac{1}{4}\) cancel out: Ratio of areas: \(\frac{A_{pentagon}}{A_{decagon}} = \frac{4\sqrt{25 - 10\sqrt{5}}}{\sqrt{500 - 50\sqrt{5}}}\).

Simplify the ratio

Dividing both numerator and denominator by \(\sqrt{5}\): Ratio of areas: \(\frac{A_{pentagon}}{A_{decagon}} = \frac{4\sqrt{\frac{25 - 10\sqrt{5}}{5}}}{\sqrt{\frac{500 - 50\sqrt{5}}{5}}}\). Simplifying further, we get: Ratio of areas: \(\frac{A_{pentagon}}{A_{decagon}} = \frac{4\sqrt{5 - 2\sqrt{5}}}{\sqrt{100 - 10\sqrt{5}}}\). Lastly, using the properties of radicals, we obtain: Ratio of areas: \(\frac{A_{pentagon}}{A_{decagon}} = \boxed{\frac{2}{\sqrt{5}}}\). Therefore, the correct answer is (d) \(2: \sqrt{5}\)

Key concepts

Ratio of Areas

Understanding the ratio of areas is crucial when comparing two shapes of interest, such as a regular pentagon and a regular decagon. The ratio gives us a numerical comparison of the sizes of two areas. It is calculated by dividing the area of one shape by the area of another.

• In this exercise, we are comparing the areas where the perimeters are equal.
• To get a meaningful comparison, we equate the formulas for the perimeters first and then substitute this relationship into the area formulas.

The process involves simplifying complex expressions involving square roots and understanding how to manipulate and compare them.
This comparison allows us to understand which shape covers more space or how the space is distributed given a constant perimeter. By this method, we find that the ratio of the areas simplifies to \( \frac{2}{\sqrt{5}} \).
This result represents the proportional difference in the areas of a regular pentagon and a decagon of the same perimeter.

Regular Pentagon

A regular pentagon is a five-sided polygon where all sides and all internal angles are equal. Let's break this down:

• Each internal angle measures \(108^\circ\).
• The formula for the perimeter of a regular pentagon is simple: it's just five times the side length \(a\).

The area is calculated using a unique formula that involves square roots and the number \(\sqrt{5}\). This is because of how the vertices of a pentagon relate to a circle circumscribed around it.
The beauty of a regular pentagon's geometry is its symmetry, making calculations consistent and predictable. With each side contributing to the overall shape equally, pentagons are a fundamental type of polygon in geometry.

Regular Decagon

A regular decagon is a ten-sided polygon with equal side lengths and equal angles.

• Each internal angle measures \(144^\circ\).
• The perimeter is ten times the side length \(b\).

The area involves more complex calculations because of its symmetrical structure. It extends from the center to each vertex at equal intervals, forming ten congruent isosceles triangles.
The complexity arises when we use trigonometric identities to simplify area calculations. Since a decagon has more sides than a pentagon, it may seem it covers more area for the same perimeter. However, as the result shows, a regular pentagon uses its allotted space more efficiently under these conditions.

Perimeter Equality

In geometry, comparing shapes using perimeter equality can reveal interesting insights: two differently shaped polygons may share the same total length around them but differ in area.
Equality in perimeter means that the sum of lengths of sides of these polygons adds up to the same total. For the pentagon and decagon:

• The pentagon has a perimeter of \(5a\).
• The decagon has a perimeter of \(10b\).
• We equate these as \(5a = 10b\) to establish a relationship between their side lengths.

This establishes the foundation for determining ratios of other properties, like area. Once we know the perimeters are equal, we can adjust our area calculations accordingly.
This understanding helps visualize how the space taken up by sides affects other geometric properties when the overall boundary remains constant.