A circle is one of the most widely recognizable geometric shapes, but exploring the mathematical concepts of diameter and area can sometimes feel tricky. Whether you are measuring the size of round rug you need to purchase or determining the space you need to construct a round garden or patio, knowing how to calculate the area of a circle from its diameter is a valuable skill. Show TL;DR (Too Long; Didn't Read)The area of a circle is the amount of space the circle covers. The formula for calculating the area of a circle is A = πr2 where pi (π) equals 3.14 and the radius (r) is half the diameter. The first step for calculating the area of a circle from its diameter is to find that diameter. While math problems often list this value, in the real world, you must find the diameter yourself. The diameter is the length of a line that begins at the edge of the circle, passes through the center of the circle, and ends at the opposite edge of the circle. To measure, you will need a ruler for small circles or a tape measure for large circles. Once you have the diameter (d) of the circle, you can find the radius (r) using the equation d=2r. The radius of a circle is the distance from the center of the circle to any point on the edge of the circle. The radius is also half of the diameter. If your diameter is a simple number, you can likely calculate the radius in your head. If not, rearrange the equation to find for r r = \frac{d}{2} and solve. You are now ready to use the equation for area: A = πr^2 Pi (π) is a non-algebraic number that represents the ratio of the distance around the circle (circumference) to its diameter, usually estimated as 3.14. To solve for area, square the radius (radius times radius) then multiply by 3.14. Since area is a measure of two dimensions, you always report area in square units like square inches (in2) or square feet (ft2). This is especially important when calculating the area of a circle for an assignment since an answer without correctly reported units is likely incorrect or incomplete. Any time you need to determine the space inside a circle or the amount of space a circle covers, you can use the equation for the area of a circle. Especially for real world applications of this skill, measuring diameter is often the simplest way to start. Table Of Contents
What Is The Area Of A Circle?A circle is not a square, but a circle's area (the amount of interior space enclosed by the circle) is measured in square units. Finding the area of a square is easy: length times width. A circle, though, has only a diameter, or distance across. It has no clearly visible length and width, since a circle (by definition) is the set of all points equidistant from a given point at the center. Yet, with just the diameter, or half the diameter (the radius), or even only the circumference (the distance around), you can calculate the area of any circle. How To Find The Area Of A CircleRecall that the relationship between the circumference of a circle and its diameter is always the same ratio, 3.14159265, pi, or π. That number, π, times the square of the circle's radius gives you the area of the inside of the circle, in square units. Area Of A Circle FormulaIf you know the radius, r, in whatever measurement units (mm, cm, m, inches, feet, and so on), use the formula π r2 to find area, A: The answer will be square units of the linear units, such as mm2, cm2, m2, square inches, square feet, and so on. Here is a circle with a radius of 7 meters. What is its area? [insert drawing of 14-m-wide circle, with radius labeled 7 m] A = π ·r2 A = π × 72 A = π × 49 A = 153.9380 m2 Area Of A Circle Using DiameterIf you know the diameter, d, in whatever measurement units, take half the diameter to get the radius, r, in the same units. Here is the real estate development of Sun City, Arizona, a circular town with a diameter of 1.07 kilometers. What is the area of Sun City?  First, find half the diameter, given, to get the radius: 1.07 2 = 0.535 km = 535 m Plug in the radius into our formula: A = π·r2 A = π × 5352 A = π × 286,225 A = 899,202.3572 m2 To convert square meters, m2, to square kilometers, km2, divide by 1,000,000: A = 0.8992 km2 Sun City's westernmost circular housing development has an area of nearly 1 square kilometer! How to Calculate the Area of a CircleTry these area calculations for four different circles. Be careful; some give the radius, r, and some give the diameter, d. Remember to take half the diameter to find the radius before squaring the radius and multiplying by π. Problems
Do not peek at the answers until you do your calculations! Answers
A = πr2 A = π × 203 mm2 A = 637.7433 mm2 A = πr2 A = π × 60 m2 A = 188.4955 m2 A = πr2 A = π × 13 in2 A = 530.9291 in2 A = πr2 A = π × 736 in2 A = 1,701,788.17 in2 That is 11,817.97 ft2 of pizza! Yum! Anyway, how did you do on the four problems? Area Of A Circle Using CircumferenceIf you have no idea what the radius or diameter is, but you know the circumference of the circle, C, you can still find the area. Area and Circumference FormulaCircumference (the distance around the circle) is found with this formula: C = 2πr That means we can take the circumference formula and "solve for r," which gives us: r = C2π We can replace r in our original formula with that new expression: A = π C2π2 That expression simplifies to this: That formula works every time! How To Find The Area With CircumferenceHere is a beautiful, reasonable-sized pizza you and three friends can share. You happen to know the circumference of your pizza is 50.2655 inches, but you do not know its total area. You want to know how many square inches of pizza you will each enjoy. [insert cartoon drawing of typical 16-inch pizza but do not label diameter] Substitute 50.2655 inches for C in the formula: A = 50.265524π A = 2,526.62044π A = 201.0620 in2 Equally divide that total area for a full-sized pizza among four friends, and you each get 50.2655 in2 of pizza! That's about a third of a square foot for each of you! Yum, yum! Next Lesson:Area of a Sector of a Circle |
How do you find the area in a circle
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