In math (especially geometry) and science, you will often need to calculate the surface area, volume, or perimeter of a variety of shapes. Whether it’s a sphere or a circle, a rectangle or a cube, a pyramid or a triangle, each shape has specific formulas that you must follow to get the correct measurements.
We’re going to examine the formulas you will need to figure out the surface area and volume of three-dimensional shapes as well as the area and perimeter of two-dimensional shapes. You can study this lesson to learn each formula, then keep it around for a quick reference next time you need it. The good news is that each formula uses many of the same basic measurements, so learning each new one gets a little easier.
Surface Area and Volume of a Sphere
A three-dimensional circle is known as a sphere. In order to calculate either the surface area or the volume of a sphere, you need to know the radius (r). The radius is the distance from the center of the sphere to the edge and it is always the same, no matter which points on the sphere’s edge you measure from.
Once you have the radius, the formulas are rather simple to remember. Just as with the circumference of the circle, you will need to use pi (π). Generally, you can round this infinite number to 3.14 or 3.14159 (the accepted fraction is 22/7).
- Surface Area = 4πr2
- Volume = 4/3 πr3
Surface Area and Volume of a Cone
A cone is a pyramid with a circular base that has sloping sides which meet at a central point. In order to calculate its surface area or volume, you must know the radius of the base and the length of the side.
If you do not know it, you can find the side length (s) using the radius (r) and the cone’s height (h).
- s = √(r2 + h2)
With that, you can then find the total surface area, which is the sum of the area of the base and area of the side.
- Area of Base: πr2
- Area of Side: πrs
- Total Surface Area = πr2 + πrs
To find the volume of a sphere, you only need the radius and the height.
- Volume = 1/3 πr2h
Surface Area and Volume of a Cylinder
You will find that a cylinder is much easier to work with than a cone. This shape has a circular base and straight, parallel sides. This means that in order to find its surface area or volume, you only need the radius (r) and height (h).
However, you must also factor in that there is both a top and a bottom, which is why the radius must be multiplied by two for the surface area.
- Surface Area = 2πr2 + 2πrh
- Volume = πr2h
Surface Area and Volume of a Rectangular Prism
A rectangular in three dimensions becomes a rectangular prism (or a box). When all sides are of equal dimensions, it becomes a cube. Either way, finding the surface area and the volume require the same formulas.
For these, you will need to know the length (l), the height (h), and the width (w). With a cube, all three will be the same.
- Surface Area = 2(lh) + 2(lw) + 2(wh)
- Volume = lhw
Surface Area and Volume of a Pyramid
A pyramid with a square base and faces made of equilateral triangles is relatively easy to work with.
You will need to know the measurement for one length of the base (b). The height (h) is the distance from the base to the center point of the pyramid. The side (s) is the length of one face of the pyramid, from the base to the top point.
- Surface Area = 2bs + b2
- Volume = 1/3 b2h
Another way to calculate this is to use the perimeter (P) and the area (A) of the base shape. This can be used on a pyramid that has a rectangular rather than a square base.
- Surface Area = ( ½ x P x s ) + A
- Volume = 1/3 Ah
Surface Area and Volume of a Prism
When you switch from a pyramid to an isosceles triangular prism, you must also factor in the length (l) of the shape. Remember the abbreviations for base (b), height (h), and side (s) because they are needed for these calculations.
- Surface Area = bh + 2ls + lb
- Volume = 1/2 (bh)l
Yet, a prism can be any stack of shapes. If you have to determine the area or volume of an odd prism, you can rely on the area (A) and the perimeter (P) of the base shape. Many times, this formula will use the height of the prism, or depth (d), rather than the length (l), though you may see either abbreviation.
- Surface Area = 2A + Pd
- Volume = Ad
Area of a Circle Sector
The area of a sector of a circle can be calculated by degrees (or radians as is used more often in calculus). For this, you will need the radius (r), pi (π), and the central angle (θ).
- Area = θ/2 r2 (in radians)
- Area = θ/360 πr2 (in degrees)
Area of an Ellipse
An ellipse is also called an oval and it is, essentially, an elongated circle. The distances from the center point to the side are not constant, which does make the formula for finding its area a little tricky.
To use this formula, you must know:
- Semiminor Axis (a): The shortest distance between the center point and the edge.
- Semimajor Axis (b): The longest distance between the center point and the edge.
The sum of these two points does remain constant. That is why we can use the following formula to calculate the area of any ellipse.
- Area = πab
On occasion, you may see this formula written with r1 (radius 1 or semiminor axis) and r2 (radius 2 or semimajor axis) rather than a and b.
- Area = πr1r2
Area and Perimeter of a Triangle
The triangle is one of the simplest shapes and calculating the perimeter of this three-sided form is rather easy. You will need to know the lengths of all three sides (a, b, c) to measure the full perimeter.
- Perimeter = a + b + c
To find out the triangle’s area, you will need only the length of the base (b) and the height (h), which is measured from the base to the peak of the triangle. This formula works for any triangle, no matter if the sides are equal or not.
- Area = 1/2 bh
Area and Circumference of a Circle
Similar to a sphere, you will need to know the radius (r) of a circle to find out its diameter (d) and circumference (c). Keep in mind that a circle is an ellipse that has an equal distance from the center point to every side (the radius), so it does not matter where on the edge you measure to.
- Diameter (d) = 2r
- Circumference (c) = πd or 2πr
These two measurements are used in a formula to calculate the circle’s area. It’s also important to remember that the ratio between a circle’s circumference and its diameter is equal to pi (π).
- Area = πr2
Area and Perimeter of a Parallelogram
The parallelogram has two sets of opposite sides that run parallel to one another. The shape is a quadrangle, so it has four sides: two sides of one length (a) and two sides of another length (b).
To find out the perimeter of any parallelogram, use this simple formula:
- Perimeter = 2a + 2b
When you need to find the area of a parallelogram, you will need the height (h). This is the distance between two parallel sides. The base (b) is also required and this is the length of one of the sides.
- Area = b x h
Keep in mind that the b in the area formula is not the same as the b in the perimeter formula. You can use any of the sides—which were paired as a and b when calculating perimeter—though most often we use a side that is perpendicular to the height.
Area and Perimeter of a Rectangle
The rectangle is also a quadrangle. Unlike the parallelogram, the interior angles are always equal to 90 degrees. Also, the sides opposite one another will always measure the same length.
To use the formulas for perimeter and area, you will need to measure the rectangle’s length (l) and its width (w).
- Perimeter = 2h + 2w
- Area = h x w
Area and Perimeter of a Square
The square is even easier than the rectangle because it is a rectangle with four equal sides. That means you only need to know the length of one side (s) in order to find its perimeter and area.
- Perimeter = 4s
- Area = s2
Area and Perimeter of a Trapezoid
The trapezoid is a quadrangle that can look like a challenge, but it’s actually quite easy. For this shape, only two sides are parallel to one another, though all four sides can be of different lengths. This means that you will need to know the length of each side (a, b1, b2, c) to find a trapezoid’s perimeter.
- Perimeter = a + b1 + b2 + c
To find the area of a trapezoid, you will also need the height (h). This is the distance between the two parallel sides.
- Area = 1/2 (b1 + b2) x h
Area and Perimeter of a Hexagon
A six-sided polygon with equal sides is a regular hexagon. The length of each side is equal to the radius (r). While it may seem like a complicated shape, calculating the perimeter is a simple matter of multiplying the radius by the six sides.
- Perimeter = 6r
Figuring out the area of a hexagon is a little more difficult and you will have to memorize this formula:
- Area = (3√3/2 )r2
Area and Perimeter of an Octagon
A regular octagon is similar to a hexagon, though this polygon has eight equal sides. To find the perimeter and area of this shape, you will need the length of one side (a).
- Perimeter = 8a
- Area = ( 2 + 2√2 )a2