Problem 63 - Algebra and Trigonometry

Question

Find the area of a sector with central angle 1 rad in a circle of radius \(10 \mathrm{m} .\)

Step-by-Step Solution

Answer

The area of the sector is 50 square meters.

Step 1: Understanding the Formula for the Area of a Sector

The area of a sector in a circle with radius \( r \) and central angle \( \theta \) (in radians) is given by \( A = \frac{1}{2} r^2 \theta \). This formula helps us find the portion of the circle's area occupied by the sector.

Step 2: Substituting Known Values

Given that the radius \( r = 10 \) meters and the central angle \( \theta = 1 \) radian, we substitute these values into the formula: \( A = \frac{1}{2} \times 10^2 \times 1 \).

Step 3: Simplifying the Expression

Calculate the value: \( A = \frac{1}{2} \times 100 \times 1 = 50 \). Thus, the area of the sector is 50 square meters.

Key Concepts

Circle Geometry

Circle geometry deals with properties and definitions related to circles. A circle is a set of points that are all the same distance from a central point. This distance is known as the radius.
A circle's geometry encompasses concepts such as diameters, chords, and arcs.

• The diameter is the longest distance across the circle, equal to twice the radius.
• A chord is a line segment with its endpoints on the circle.
• An arc is a portion of the circle's circumference.

A sector, which we are interested in here, is a slice of the circle—like a 'pizza slice.' It includes the arc along with the two radii that form an angle at the circle's center.

Central Angle

The central angle is a key concept in circle geometry. It is formed by two radii and the arc they encompass. The central angle measures how 'broad' the sector is. In our example, the central angle is given as 1 radian. This angle tells us how large a section of the circle's circumference is being considered. The central angle is crucial because it determines the size of the sector. When solving problems involving sectors, you'll often use the central angle to compute the area, as it directly influences the sector's fraction of the whole circle.

Radians

Radians are a unit of angular measure used in many areas of mathematics and geometry. Unlike degrees, which split a circle into 360 parts, radians relate directly to the radius of the circle.

• Your typical full circle has an angle of \(2\pi\) radians, approximately 6.28 radians.
• One radian is the angle created when the arc length equals the circle's radius.

Using radians simplifies many mathematical formulas, especially in trigonometry and calculus. In calculating a sector's area, the angle in radians directly multiplies with the radius squared, simplifying to the area formula: \(A = \frac{1}{2} \times r^2 \times \theta\).

Radius

The radius of a circle is the distance from its center to any point on its boundary. It is a fundamental element of circle geometry, as it defines the circle's size. All radii in a circle are identical, maintaining the circle's symmetry and uniformity. In the sector's area formula, the radius is squared, showcasing its importance in determining the sector's size. With a radius of 10 meters in our example, the squared radius is 100. When multiplied by the central angle in radians, it helps find the area the sector occupies within the circle. This foundational property of the circle influences the computations related to both its perimeter and the sector's area.