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A **semicircle**, as the name suggests, is one-half of a circle. While circles have been central to human civilization for millennia, from the wheels of carts to the intricate designs in cathedrals, **semicircles** too have their unique charm and utility.

Understanding the **equation** of a **semicircle** offers a glimpse into a world where mathematics marries art, architecture, and more.

**Defining the Equation of a Semicircle**

The equation of a **semicircle **is defined as:

**Upper Semicircle**: y = $\sqrt{r^2 – x^2}$**Lower Semicircle**: y = – $\sqrt{p^2 – x^2}$

Depending on its orientation (upper or lower half), a **semicircle** has an **equation** governed by the above two equations.

**Defining a Semicircle**

A **semicircle** can be defined as the shape that results when you cut a full circle along its diameter. The cut divides the circle into two congruent shapes, each being a **semicircle.** The defining feature of a **semicircle** is its diameter, which doubles as its straight edge, unlike a circle, which is defined by its center and radius.

Figure-1.

**Mathematical Representation **

The **equation** of a full circle with a center at the origin (0, 0) of a coordinate plane and a radius $r$ is $x^2$ $+ $y^2$= $r^2$$. However, a **semicircle,** depending on its orientation (upper or lower half), has an **equation** governed by the following:

**Upper Semicircle**: y = $\sqrt{r^2 – x^2}$**Lower Semicircle**: y = – $\sqrt{p^2 – x^2}$

If the **semicircle’s** center is not at the origin and is instead $(h,k)$, then the **equation** modifies accordingly.

**Adjusting the Center**

While the standard **equation** of a **semicircle** assumes the center is at the origin, in many cases, it’s not. Let’s say the center of our **semicircle** is at $(h,k)$. The **equation** for the upper half becomes:

$y−k= $x^2$− $\sqrt{x-h}$$

For the lower half, it is:

$y−k= – $r^2$− $\sqrt{x-h}$$

**Exercise **

### Example 1

Find the **equation** of the upper **semicircle** with radius 4 units and centered at the origin.

Figure-2.

### Solution

**Equation** of the circle:

$x^2$ $+ $y^2$=16$

For the upper **semicircle**:

y = $\sqrt{16 – x^2}$

### Example 2

Find the **equation** of the lower **semicircle** with a radius of 3 units centered at the origin.

Figure-3.

### Solution

**Equation** of the circle:

$x^2$ $+ $y^2$= 9$

For the lower **semicircle**:

$y= – $\sqrt{9 – x^2}$$

### Example 3

Determine the **equation** of the upper **semicircle** with radius 5 and centered at (2, 3).

### Solution

**Equation** of the circle:

${(x-2)^2 – (y – 3)^2}$ = 25

For the upper **semicircle**:

y = 3 + $\sqrt{25 – (x-2)^2}$

## Applications of the Equation of a Semicircle in Different Fields

**Architecture & Construction****Arches**: One of the most iconic applications of**semicircles**in architecture is the semicircular arch, prevalent in Roman and Gothic architectural designs. These arches are not only aesthetically pleasing but also distribute weight efficiently, making structures more stable.**Domes**: Semi-domes, which are essentially 3D adaptations of**semicircles**, are often found in religious and monumental architecture. Their shape provides both structural stability and acoustic benefits.**Design Elements**: Semicircular patios, terraces, and balconies are popular in modern designs, offering unique vistas and efficient use of space.

**Art & Design****Patterns and Motifs**: Semicircular patterns are popular in various art forms, from mosaics to fabric designs, offering symmetrical aesthetics.**Sculptures**: Many sculptures, especially fountains and monuments, utilize semicircular designs for visual balance and focus.

**Engineering****Bridges**: Some bridges employ semicircular designs for their arches, distributing the forces of tension and compression efficiently.**Tunnels**: The cross-sectional design of many tunnels is semicircular, which helps in optimizing structural integrity against the forces of the earth.

**Mathematics & Physics****Trigonometry**: The unit circle’s representation in trigonometry often involves using**semicircles**to explain sine, cosine, and their relationships.**Optics**: The principle of reflection in concave mirrors (which are semicircular in shape) is based on the**equation**of the**semicircle**.

**Nature & Environment****Landscaping**: Gardens and parks sometimes employ semicircular designs for aesthetic appeal, especially in water features or seating arrangements.**Environmental Studies**: The study of phenomena like rainbows and horizons, which can often appear semicircular, involves understanding the properties of this shape.

**Sports****Tracks**: Many athletic tracks have semicircular ends, and understanding the curvature helps in designing the lanes.**Fields**: Certain sports, like basketball, involve semicircular regions (e.g., the three-point arc).

**Technology & Electronics****Communication**: Satellite dishes, which are often semicircular, utilize the shape’s properties to capture signals effectively.**Device Design**: Many gadgets and electronics, from headphones to certain screen designs, employ semicircular aesthetics and functionality.

**Medicine****Medical Imaging**: Devices like MRIs have a semicircular design, allowing the patient to be at the center of the imaging device for optimal results.**Prosthetics**: Some prosthetic designs, especially for joints, may use semicircular components to mimic the natural curvature of the body.

**Challenges & Complexities**

While the **semicircle’s** **equation** might appear straightforward, many complexities arise when:

**Intersecting with Other Shapes**

The point of intersection of a **semicircle** with other geometric figures can result in complicated **equations.**

**Transformation**

Translating, rotating, or performing other transformations on a **semicircle** can introduce complexities in the mathematical representation.

**Real-World Imperfections**

When transitioning from paper to reality, imperfections, and challenges, like material constraints or environmental factors, can influence the ideal semicircular shape.

*All images were created with GeoGebra.*