How to Calculate the Area of 2D and 3D Shapes

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Learn how to calculate the area of simple and complex shapes

Co-authored by David Jia

Last Updated: December 29, 2023 Fact Checked

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Finding the area of an object is easy as long as you understand the technique and formulas involved. If you have the right knowledge, you can find the area and surface area of any given object. Read on to learn the proper formulas for both 2D and 3D objects.

Things You Should Know

Find the area of an object by multiplying its width by its length.
Use more specific formulas to calculate the area of more specific shapes. For example, the formula for finding a triangle's area is length x width x 1/2.
Calculate the surface area of a 3D object by using its specified formula. For example, the formula for finding a cone's surface area is π x r x s + πr².

Section 1 of 2:

How to Find the Area of 2D Shapes

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1
Break down the object into smaller shapes. If you're not working with a simple easily-identifiable shape, like a circle or a trapezoid, then you may be working with a shape that is made up of several other shapes. You have to recognize what those are so you can break the larger object down into a series of smaller objects.^{[1]
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- In this case, this object is composed of the following shapes: a triangle, a trapezoid, a rectangle, a square, and a semi-circle (half of a circle).
2
Write down the formulas for finding the area of each of these shapes. These formulas will allow you to use the given measurements of each shape to find their areas. Here are the formulas for finding the area of each shape:^{[2]
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- Area of a Square = side² = a²
- Area of a Rectangle = width x height = w x h
- Area of a Trapezoid = [(side 1 + side 2) x height]/2 = [(a + b) x h]/2
- Area of a Triangle = base x height x 1/2 = (b x h)/2
- Area of a Semi-Circle = (π x radius²)/2 = (π x r²)/2
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3
Write down the dimensions of each shape. Once you write down the formulas, write down the dimensions of each shape so you can plug them in. Here are the dimensions of each shape:
- Square: a = 2.5 in
- Rectangle = w = 4.5 in, h = 2.5 in
- Trapezoid = a = 3 in, b = 5 in, h = 5 in
- Triangle = b = 3 in, h = 2.5 in
- Semi-Circle = r = 1.5 in
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Use the formulas and dimensions to find the area of each object and add them up. Finding the area of each shape will lead you to find the area of each part of the shape; once you've found the area of each shape using the formula and measurements you were given, all you have to do is add up each area to find the area of the entire object. When calculating area, you have to remember to state the area in square units. The area of the entire object is 44.78 in². Here's how you get it:^{[3]
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- Find the area of each shape:
  - Area of Square = 2.5 in² = 6.25 in²
  - Rectangle = 4.5 in x 2.5 in = 11.25 in²
  - Trapezoid = [(3 in + 5 in) x 5 in]/2 = 20 in²
  - Triangle = 3 in x 2.5 in x 1/2 = 3.75 in²
  - Semi-Circle = 1.5 in² x π x 1/2 = 3.53 in²
- Add the areas of each shape together:
  - Area of Object = Area of Square + Area of Rectangle + Area of Trapezoid + Area of Semi-Circle
  - Area of Object = 6.25 in² + 11.25 in² + 20 in² + 3.75 in² + 3.53 in²
  - Area of Object = 44.78 in²

Section 2 of 2:

How to Find the Surface Area of 3-D Objects

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1
Write down the formulas for finding the surface area of each shape. The surface area is the total area of an object's faces and curved surfaces. Every three dimensional object has a surface area; the volume is the amount of space taken up by the object. Here are the formulas for finding the surface area of a variety of objects:^{[4]
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- Surface area of a square = 6 x side² = 6s²
- Surface area of a cone = π x radius x side + π x radius² = π x r x s + πr²
- Surface area of a sphere = 4 x π x radius² = 4πr²
- Surface area of a cylinder = 2 x π x radius² + 2 x π x radius x height = 2πr² + 2πrh
- Surface area of a square-based pyramid = side of base² + 2 x side of base x h = b² + 2bh
2
Write down the dimensions of each shape. Here they are:
- Cube = side = 3.5 in
- Cone = r = 2 in, h = 4 in
- Sphere = r = 3 in
- Cylinder = r = 2 in, h = 3.5 in
- Square-Based Pyramid = b = 2 in, h = 4 in
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Calculate the surface area of each shape. Now, all you have to do is plug the dimensions of each shape into the formula for finding the surface area of each shape and you're all done. Here's how you do it:^{[5]
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- Surface area of the cube = 6 x 3.5² = 73.5 in²
- Surface area of the cone = π(2 x 4) + π x 2² = 37.7 in²
- Surface are of the sphere = 4 x π x 3² = 113.09 in²
- Surface are of the cylinder = 2π x 2² + 2π(2 x 3.5) = 69.1 in²
- Surface area of the square-based pyramid = 2² + 2(2 x 4) = 20 in²

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How can I calculate the volume of a 3D shape?

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All shapes are different, but here is how to calculate cubes specifically: First you would need to have all of the dimensions (length, width, and height) of the cube. Then, you would multiply the dimensions and the formula would be l*w*h or length times width times height in word form.

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How do I calculate area given length and width?

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In the case of a square, rectangle, or parallelogram, multiply length by width.

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If a line is 100 meters in length, is the area constant for any object it forms; triangle, square, circle?

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No, the area varies by shape. For all shapes of a given perimeter, the circle has the largest area.

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Tips

Measure dimensions of the chief objects using a scale or vernier calipers.

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Warnings

Don't confuse between area and surface area both are the same but are used differently. Area is used in case of planar objects and surface area is used in case of 3D objects.

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