Question

Sketch two polygons that both have a perimeter of 12 units, but that have different areas.

Short answer

Equilateral triangle and a 3x3 rectangle both have a perimeter of 12 but different areas.

Step-by-step solution

Define the Goal

We are tasked with sketching two polygons, each having a perimeter of 12 units, but with different areas. This involves considering different shapes and calculations based on their properties.

Choose the Polygons

Let's select two simple polygons for demonstration – a triangle and a rectangle. These shapes can be more easily manipulated to have the same perimeter yet different configurations and thus areas.

Calculate Triangle Perimeter

For the triangle, choose an equilateral triangle with each side measuring 4 units. Since all sides are equal, the perimeter is calculated as: \[ 3(x) = 3(4) = 12 \] units, which satisfies the perimeter requirement.

Calculate Triangle Area

To calculate the area of an equilateral triangle with side length 4, we use the formula: \[ \text{Area} = \frac{\sqrt{3}}{4} \times s^2 = \frac{\sqrt{3}}{4} \times 4^2 = 4\sqrt{3} \] square units.

Calculate Rectangle Perimeter

For the rectangle, assign a length of 3 units and a width of 3 units. The perimeter is then calculated as: \[ 2(l + w) = 2(3 + 3) = 12 \] units, satisfying the perimeter requirement.

Calculate Rectangle Area

For the area of the rectangle with length 3 and width 3, use the formula: \[ \text{Area} = l \times w = 3 \times 3 = 9 \] square units.

Compare Areas

The equilateral triangle has an area of \( 4\sqrt{3} \) square units, while the rectangle has an area of 9 square units. These are indeed different, confirming different configurations can yield different areas even with the same perimeter.

Key concepts

polygon properties

Polygons are flat, two-dimensional shapes with straight sides. The key characteristic of a polygon is that it must have at least three sides and they must form a closed shape.

• Examples include triangles, rectangles, pentagons, and hexagons.
• Polygons are named based on the number of sides they have: a triangle has 3 sides, a rectangle or quadrilateral has 4 sides, etc.
• Convex polygons have all interior angles less than 180°, whereas concave polygons have at least one angle greater than 180°.
• Each polygon side is connected to another, forming vertices or corners.

Understanding properties like symmetry, congruence, and equilateral (all sides equal) or equiangular (all angles equal) nature helps in solving geometric problems. In this task, we focus on equilateral triangles and rectangles, each having unique properties affecting their areas and perimeters.

perimeter and area

Perimeter and area are fundamental concepts in geometry that deal with the measurement of space.
- **Perimeter** is the total length around a polygon. It is calculated by adding the length of all its sides: - For an equilateral triangle with a side length of 4 units, the perimeter is \( 4 + 4 + 4 = 12 \) units. - For a rectangle with lengths 3 and 3 units for two sides, the perimeter is \( 3 + 3 + 3 + 3 = 12 \) units.- **Area** represents the amount of two-dimensional space inside a polygon. - The formula for an equilateral triangle’s area is \( \frac{\sqrt{3}}{4} \times s^2 \), where \( s \) is the side length. - The area of a rectangle is straightforward to calculate, using \( l \times w \), where \( l \) is length and \( w \) is width. Knowing how to calculate perimeter and area is essential for comparing different polygons and understanding how their sizes change even if their perimeter remains constant.

geometry calculations

Geometry calculations often require the use of specific formulas tailored to the properties of shapes involved.
When calculating the area and perimeter, the formulas depend on the shape.

• **Equilateral Triangle:** For triangles, especially equilateral ones, using the Pythagorean theorem can be helpful in understanding the relationships between the sides and height. For our calculation:
  • The formula \( \text{Area} = \frac{\sqrt{3}}{4} \times s^2 \) is derived using basic trigonometry.
  • For side length 4, it translates to \( 4\sqrt{3} \) square units.
• **Rectangle:** The calculations are more simplistic.
  • Use \( \text{Area} = l \times w \), which gives us 9 square units for a rectangle with equal side lengths of 3.

It’s crucial to choose suitable shapes for specific problems as demonstrated, where both ask for the same perimeter but result in varying areas, showcasing geometry's intricate nature. Practicing with different polygons helps strengthen the understanding of these calculations.