Ever noticed how a circle, when sliced cleanly down the middle through its diameter, gives you two perfect halves? That shape you get is called a semicircle! It’s basically half of a circle—nothing too tricky about it. Think of it like folding a circular sheet of paper exactly in half; both sides match up perfectly. You’ve probably seen semicircles all the time without even realizing it—like those clear plastic protractors you use in geometry class to measure angles or draw arcs. Yep, those are semicircle-shaped!

In this article, we're going to break down everything you need to know about semicircles—not just their definition, but also all the essential formulas tied to them, like area, perimeter, and more. Plus, we'll talk about symmetry in semicircles (spoiler alert: they’re super symmetrical!) and walk through some step-by-step solved examples to make sure you really get it. Whether you're prepping for the SAT, ACT, PSAT/NMSQT, GED, GRE, GMAT, AP Exams, PERT, Accuplacer, or even the MCAT, understanding semicircles can definitely give you an edge. So, let’s dive in and make semicircles simple and stress-free!

Semi Circle

Since a semi circle is half of a circle, the area of a semi circle will be half the area of a circle.

Definition of semi circle : As the name suggests, a semi circle is half of a circle. If an arc of a circle with its two ends on the diameter of a circle cuts the circle, the two arcs obtained are called semi circles. We can see semi-circular shapes around us in the form of a speedometer, a protractor, etc.

The figure above represents a semi circle PQR with the arc and the diameter with both the end points. Also,

• J is the center of the circle
• PJ and QJ are the radii.

Semi Circle Formula

Some of the most commonly used formulae in the case of semicircle are:

Area of a Semicircle

Area of a semi circle is the region enclosed by a semi circle.

• Since a semi circle is half of a circle, then the area of a semi circle is half that of a circle.
• As the area of a circle is \(\pi r^2\), area of a semi circle becomes \(\frac{\pi r^2}{2}\).
• Here the value of \(\pi\) is 3.14 or 22/7.

Derivation of Area of a Semi circle

Let us see how this formula has been derived.

We have already read that the area of a semi circle is half that of a circle. Also, we know that the area of a circle is number of square units inside the circle.

Step 1 : Consider the given figure, where we have embedded a polygon inside a circle. This polygon can be further broken down into n isosceles triangles. One such isosceles triangle is represented below:

Step 2 : By the definition of area of triangle, the area of the given isosceles triangle is \(\frac{1}{2}\times h\times s\)

Step 3 : As there are n number of triangles in a polygon, so the area of a polygon becomes = \(\frac{1}{2}\times h\times s\times n\)

Step 4 : Also, \(n\times s\) is the perimeter of a polygon. As the polygon goes on looking more of a circle, this value approaches to the circumference of a circle that is \(2\pi r\)

Step 5 : So, we can substitute \(2\pi r\) for \(n\times s\) , we get Area of the polygon = \(\frac{h}{2}\left(2\pi r\right)\)

Step 6 : Moreover, as the number of sides of a polygon increases, the triangle gets narrower, so as s approaches zero, h and r have the same length.

Step 7 : We can substitute r for h,

Step 8 : Area of the polygon = \(\frac{h}{2}\left(2\pi r\right)\) =\(\frac{\left(2\times r\times r\times\pi\right)}{2}\)

Step 9 : On further rearranging, we get, Area =\(\pi r^2\)

We know that area of a semi circle is half the area of a circle, so, area of a semi circle is: \(\frac{\pi r^2}{2}\)

Learn about Secant of a Circle and Concentric Circles

Perimeter of a Semicircle

The perimeter of a semi circle is defined as the length of its boundary.

• As a semi circle is half that of a circle, so its perimeter is half the circumference of a circle along with the diameter of a circle.
• So, the perimeter of a semi circle becomes =\(\frac{2\pi r}{2}+d\)
• Where, r is the radius of the circle and d is the diameter of the circle.
• Also, it can be written as \(\frac{\pi d}{2}\ +\ d\), where d is the diameter.

Learn about Circular Permutation

Derivation of Perimeter of a Semicircle

Let us check how it is derived:

Step 1 : Consider the above semi circle AMB with

Where

• AB is the diameter
• O as center.

Step 2 : We know that the circumference of a circle is \(2\pi r\).

Step 3 : Half of the circumference of the circle is \(\pi r\) or \(\frac{\pi d}{2}\).

Step 4 : Length of the diameter is d or 2r.

Step 5 : So, the perimeter of a semi circle in terms of radius is \(\frac{2\pi r}{2}+2r\)

=\(\pi r+2r\ =r\left(\pi+2\right)\)

The perimeter of a semi circle in terms of diameter is \(\frac{\pi d}{2}+d\)

Learn about Difference Between Circle and Sphere

Circumference of a Semicircle

The circumference of a semicircle is equal to half the circumference of the circle of which it is a part of circle.

• It is defined as the measurement of the arc forming a semi circle, and hence does not include the length of the diameter.
• As we know that the circumference of a circle is \(2\pi r\).
• So, the circumference of a semi circle becomes \(\pi r\).

Angle Inscribed in a Semicircle

The angle inscribed in a semi circle is always 90-degree no matter the radius of the semi circle.

In the figure above, AC is the diameter of the semicircle. The inscribed angle of the semi circle is formed by drawing a line from each end of the diameter to any point on the circumference of the semi circle. The angle thus formed is 90-degree. In this case, angle ABC is 90-degree.

Centroid of a Semicircle

A point that is in the exact center of a figure or shape is known as the centroid of that shape. In the case of a semi circle, as the center of the shape is not obvious, the centroid can be found using a formula.

The centroid of a semi circle is at a distance of \(\frac{4r}{3\pi}\) from the base of the semi circle.

Steps to find Centroid of a Semicircle

Let us check how we can find the centroid of a semi circle:

Step 1 : Let us consider the radius of a semi circle as ‘r’. This radius extends from the center of the circle to either side of the circle. Now that we know the radius, we use ‘x’ and ‘y’ coordinates to find the centroid of the semi circle using the x axis and y axis.

Step 2 : If we consider the coordinate system to be placed at the centre of the semi circle, ‘x’ coordinate will become zero. In order to find the ‘y’ coordinate we use the formula \(y= \frac{4r}{3\pi}\)

For example: With a semi circle of radius 3 cm and the coordinate system is placed in the center of the semicircle. We can use the formula to find the centroid.

\(y= \frac{4\times 3}{3\pi}\)

So, we can say that the centroid for this semi circle is found at the coordinates (0,1.27).

Semicircle Line of Symmetry

A semi circle has only one line of symmetry passing through its center. It is basically a line that divides the semi circle into two equal halves.

In the above figure, OY is the line of symmetry of the semi circle with radius ‘r’.

Properties of Semicircle

Some of the most important properties of a semi circle are listed below:

• A semi circle is a two-dimensional figure.
• It cannot be considered as a polygon as it has curve edge.
• A semi circle has a curved edge and a diameter that together make the circumference of the semi circle.
• A semi circle is exact half of a circle. So, the diameter of a circle and the two semi circles formed out of it are same.
• The area of a semi circle is half the area of a circle.

Solved Examples on Semicircle

Que 1: Neena has a circular garden behind my residence with a diameter of 21 m. Neena wants to organize a party in the backyard and want to use half of the garden. What is the area of the garden she wishes to use?

Solution 1: Radius of the circular garden = 21m

Area of the garden =\(2\pi r\)

=\(2\pi 21\)

=\(2\times 22\times 3\)

=132 sq. m.

She wants to use half the area. So, the area to be used is 132/2 sq.m = 66 sq. m

Que 2: What is the perimeter of a semi circle when the diameter is 7cm.

Solution 2: Given that the diameter of a semi circle is 7cm.

Perimeter of a semi circle in form of its diameter is = \(\frac{\pi d}{2}+d\)

=\(\frac{\frac{22}{7}\times7}{2}+7\)

=\(\frac{22}{2}+7\)

= 11 + 7

= 18 cm.

Que 3: Find the circumference of a semi circle with diameter 8 cm.

Solution 3: Diameter of a semi circle = 8cm.

Radius of a semi circle = d/2 = 4cm.

Circumference of a semi circle = \(\pi r\)

=\( 3.142\times 4\)

= 12.568 cm.

Conclusion

So, what’s the big takeaway here? A semicircle is simply half of a circle—but don’t let that simplicity fool you! Understanding how to calculate its area, perimeter, circumference, symmetry, and even its centroid gives you a strong edge in tackling geometry problems, especially on exams like the SAT, ACT, PSAT/NMSQT, GED, GRE, GMAT, AP Exams, PERT, Accuplacer, and MCAT. We broke down all the essential formulas, explained how they’re derived, and walked through examples so you can feel confident applying them. Keep practicing these concepts—you'll be acing those tricky geometry questions in no time!