What are the formulas for the perimeter and area of a trapezium, rectangle, square, parallelogram, triangle, right-angled triangle, equilateral triangle, and isosceles triangle?
Understand the Problem
The question is asking for the formulas for the perimeter and area of various geometric shapes: trapezium, rectangle, square, parallelogram, triangle (general), right-angled triangle, equilateral triangle, and isosceles triangle. This requires recalling or looking up the appropriate formulas for each shape.
Answer
Trapezium: $P = a + b + c + d$, $A = \frac{1}{2}(a + b)h$ Rectangle: $P = 2(l + w)$, $A = lw$ Square: $P = 4s$, $A = s^2$ Parallelogram: $P = 2(a + b)$, $A = bh$ Triangle: $P = a + b + c$, $A = \frac{1}{2}bh$ Right-angled triangle: $P = a + b + c$, $A = \frac{1}{2}ab$ Equilateral triangle: $P = 3a$, $A = \frac{\sqrt{3}}{4}a^2$ Isosceles triangle: $P = 2a + b$, $A = \frac{b}{2} \sqrt{a^2 - \frac{b^2}{4}}$
Answer for screen readers
Trapezium: $P = a + b + c + d$, $A = \frac{1}{2}(a + b)h$
Rectangle: $P = 2(l + w)$, $A = lw$
Square: $P = 4s$, $A = s^2$
Parallelogram: $P = 2(a + b)$, $A = bh$
Triangle: $P = a + b + c$, $A = \frac{1}{2}bh$ or $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$
Right-angled triangle: $P = a + b + c$, $A = \frac{1}{2}ab$
Equilateral triangle: $P = 3a$, $A = \frac{\sqrt{3}}{4}a^2$
Isosceles triangle: $P = 2a + b$, $A = \frac{b}{2} \sqrt{a^2 - \frac{b^2}{4}}$
Steps to Solve
- Trapezium Perimeter
The perimeter of any polygon is the sum of its sides. A trapezium has four sides, $a$, $b$, $c$, and $d$, and its perimeter is therefore:
$P = a + b + c + d$
- Trapezium Area
The area of a trapezium is given by half the sum of the parallel sides (often denoted as $a$ and $b$) multiplied by the height $h$.
$A = \frac{1}{2}(a + b)h$
- Rectangle Perimeter
A rectangle has two pairs of equal sides, length $l$ and width $w$. Thus, its perimeter is:
$P = 2l + 2w = 2(l + w)$
- Rectangle Area
The area of a rectangle is the product of its length and width:
$A = lw$
- Square Perimeter
A square has four equal sides, each of length $s$. Its perimeter is:
$P = 4s$
- Square Area
The area of a square is the side length squared:
$A = s^2$
- Parallelogram Perimeter
A parallelogram has two pairs of equal sides, $a$ and $b$. Its perimeter is:
$P = 2a + 2b = 2(a + b)$
- Parallelogram Area
The area of a parallelogram is the base $b$ multiplied by the height $h$:
$A = bh$
- General Triangle Perimeter
For any triangle with sides $a$, $b$, and $c$, the perimeter is:
$P = a + b + c$
- General Triangle Area
The area of a triangle can be found using the formula: $A = \frac{1}{2}bh$, where $b$ is the base and $h$ is the height. Alternatively, Heron's formula can be used: $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s$ is the semi-perimeter, $s = \frac{a+b+c}{2}$.
- Right-Angled Triangle Perimeter
For a right-angled triangle with sides $a$, $b$, and hypotenuse $c$, the perimeter is:
$P = a + b + c$
- Right-Angled Triangle Area
If $a$ and $b$ are the two sides forming the right angle, the area is:
$A = \frac{1}{2}ab$
- Equilateral Triangle Perimeter
An equilateral triangle has three equal sides of length $a$. Its perimeter is:
$P = 3a$
- Equilateral Triangle Area
The area of an equilateral triangle is:
$A = \frac{\sqrt{3}}{4}a^2$
- Isosceles Triangle Perimeter
An isosceles triangle has two equal sides, let's call them $a$, and a base $b$. The perimeter is:
$P = 2a + b$
- Isosceles Triangle Area
The area of an isosceles triangle can be calculated as $A = \frac{1}{2} \cdot b \cdot h$, where $b$ is the base and $h$ is the height. If we only know the sides, we can calculate the height using Pythagorean theorem realizing the height bisects the base, hence $\left(\frac{b}{2}\right)^2 + h^2 = a^2$, and $h = \sqrt{a^2 - \frac{b^2}{4}}$. Therefore, $A = \frac{b}{2} \sqrt{a^2 - \frac{b^2}{4}}$
Trapezium: $P = a + b + c + d$, $A = \frac{1}{2}(a + b)h$
Rectangle: $P = 2(l + w)$, $A = lw$
Square: $P = 4s$, $A = s^2$
Parallelogram: $P = 2(a + b)$, $A = bh$
Triangle: $P = a + b + c$, $A = \frac{1}{2}bh$ or $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$
Right-angled triangle: $P = a + b + c$, $A = \frac{1}{2}ab$
Equilateral triangle: $P = 3a$, $A = \frac{\sqrt{3}}{4}a^2$
Isosceles triangle: $P = 2a + b$, $A = \frac{b}{2} \sqrt{a^2 - \frac{b^2}{4}}$
More Information
The formulas provided are fundamental in geometry and are widely used in various fields such as engineering, architecture, and physics for calculating dimensions, areas, and perimeters of different shapes. Understanding these provides a foundation for more complex geometrical problems.
Tips
A common mistake is confusing the area and perimeter formulas. Perimeter is the distance around the shape, so it's always a sum of lengths. Area is the amount of surface the shape covers.
Another mistake is using the incorrect height in parallelograms and triangles. The height must be perpendicular to the base.
For the area of a triangle, forgetting the $1/2$ factor is common. For an Isosceles triangle, calculating the height is often a source of error.
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