Measuring Space: Perimeter and AreaClass 9 Mathematics Important Points

The total length around a circle is its circumference, given by the formula $C = 2 \pi r$ or $C = \pi d$, where $r$ is the radius and $d$ is the diameter. The value of $\pi$ is an irrational constant, approximated as $\frac{22}{7}$ or $3.14$.

The length of an arc is a fraction of the circle's circumference. The formula is $l = 2 \pi r \times \frac{\theta}{360^\circ}$, where $r$ is the radius and $\theta$ is the angle the arc subtends at the center, measured in degrees.

The area enclosed by a circle is calculated using the formula $A = \pi r^2$, where $r$ is the radius of the circle. The area is always measured in square units.

A sector is a pie-shaped part of a circle bounded by two radii and an arc. Its area is a fraction of the total circle's area, given by $A = \pi r^2 \times \frac{\theta}{360^\circ}$, where $\theta$ is the angle of the sector in degrees.

The most common formula for the area of a triangle is $A = \frac{1}{2} \times \text{base} \times \text{height}$, or $A = \frac{1}{2}bh$. The height $h$ must be the perpendicular distance from the base to the opposite vertex.

When the lengths of all three sides $a, b, c$ of a triangle are known, its area can be found using Heron's formula. First, calculate the semi-perimeter $s = \frac{a+b+c}{2}$, then the area is $A = \sqrt{s(s-a)(s-b)(s-c)}$.

The area of a parallelogram is the product of its base and its corresponding perpendicular height, given by the formula $A = b \times h$. Knowing only the side lengths is not enough to determine the area.

The area of a trapezium, which has one pair of parallel sides, is given by $A = \frac{1}{2} \times (a+b) \times h$. Here, $a$ and $b$ are the lengths of the parallel sides and $h$ is the perpendicular height between them.

A median of a triangle is a line segment from a vertex to the midpoint of the opposite side. Any median divides the triangle into two smaller triangles that have equal areas.

For a cyclic quadrilateral (a four-sided figure whose vertices all lie on a circle) with side lengths $a, b, c, d$, the area is given by Brahmagupta's formula. With semi-perimeter $s = \frac{a+b+c+d}{2}$, the area is $A = \sqrt{(s-a)(s-b)(s-c)(s-d)}$.

The perimeter of a circular sector is the total length of its boundary. It is the sum of the two radii and the arc length, calculated as $P = 2r + (2 \pi r \times \frac{\theta}{360^\circ})$.

A circular segment is the region between a chord and its arc. Its area is calculated by subtracting the area of the triangle formed by the radii and the chord from the area of the corresponding sector.

On curved running tracks, athletes in outer lanes have a longer path. To ensure all athletes run the same distance, starting positions are staggered. The stagger compensates for the difference in the circumference of the lanes.

The perimeter of any polygon is the sum of the lengths of all its sides. For a regular polygon with $n$ sides of length $a$, the perimeter is $P = n \times a$.