Back To CourseGeometry: High School
15 chapters | 160 lessons
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Amy has a master's degree in secondary education and has taught math at a public charter high school.
Pyramids are 3-dimensional objects with a polygonal base and with sides that meet at one point at the top called an apex. Looking at the pyramids of Egypt, we see that they have a square for their base, and you can definitely see that all their pyramids have just one point at the very top where all the sides meet. Because the pyramids of Egypt have a square as a base, they are called square pyramids. Yes, pyramids are named after their bases. If the base is triangular, then we can call it a triangular pyramid.
No matter what kind of pyramid you have, there are only four important measurements you need to know when it comes to finding surface area and volume.
The first is the area of the base. This we will call B. If our base is a square, then we would multiply the length and the width of the base to find the area. If our base is a triangle, then we would multiply the length and the height of the triangle and divide by two to find our base area. Some problems will give you the area of the base, while others leave it up to you to find the area. If you have to find the area, just remember to use the appropriate formula for the kind of shape you have.
The second measurement is the perimeter of the base. This we will label as P. To find the perimeter of the base, you would add up the edges of the base as you go around the base. Depending on what kind of base you have, you might have three sides, four sides or more. If the problem doesn't give you the base perimeter, then you will need to calculate it on your own by adding up all the base edges.
The third measurement is the height of the pyramid. This we can label h. The height measures how tall the pyramid stands. The last measurement is the slant length. We can label this one s. This measurement tells you the length of the side going from the top down to the edge of the base.
The formula to find the surface area - the total area of all surfaces - uses three of our measurements.
Surface Area = B + (P * s)/2
This formula tells us to multiply the base perimeter with the slant height and then to divide that by 2. We then add that value to the base area to find our answer. Let's see how this works with a sample problem. Why don't we calculate the surface area of a square pyramid similar to the ones in Egypt? For our particular pyramid, we see that each side measures 750 feet long and the pyramid has a height of 500 feet and a slant length of 650 feet.
From the information we are given, I see that we need to calculate the base area and base perimeter. Well, my base is a square, so the base area will be 750 * 750 = 562,500 sq ft. My base perimeter is 750 + 750 + 750 + 750 = 3,000 ft. So now I have my B as 562,500 sq ft, my P as 3,000 ft, my h as 500 ft and my s as 650 ft. I have all the measurements I need to proceed.
I look at my formula for surface area and I see that all I need to input are the base area, base perimeter and slant length. I go ahead and plug those numbers in where they belong: Surface Area = 562,500 + (3,000 * 650) / 2.
Then I evaluate to find my answer. I multiply my 3,000 by 650 to get 1,950,000. I divide that by 2 to get 975,000. I add this to 562,500 to get an answer of 1,537,500 feet squared. Area is always squared, so I made sure that my answer ended with my measuring units squared.
For volume - the space inside our object - the formula uses two of our measurements, the base area and the height.
Volume = (B * h)/3
The formula tells us to multiply the base area with the height and then to divide by 3. Let's see how this formula works with the pyramid that we were working on already. We've already calculated the base area and found that to be 562,500 feet squared. We know our height to be 500 feet. So we can plug these into our formula for volume: Volume = (562,500 * 500) / 3.
To get my answer, I multiply the 562,500 with the 500 to get 281,250,000, and then I divide that by 3 to get an answer of 93,750,000 feet cubed. Because volume is always cubed, my answer ends with my measuring units cubed.
We've learned that a pyramid is a 3-dimensional object with a polygonal base whose sides meet at a point called the apex at the top. The pyramids of Egypt are square pyramids because their base is a square. If the base is a triangle, then it would be called a triangular pyramid. The four important measurements that we need to find surface area and volume are base area, base perimeter, height and slant length. The slant length is the distance from the top down one of the sides to the edge of the base.
The formula for surface area - the total area of all surfaces - is Surface Area = B + (P * s) / 2. For volume - the amount of space inside our 3-dimensional object - the formula is Volume = (B * h) / 3. The B is for base area, the P for base perimeter, the h for height and the s is for slant length. To use these formulas, just plug in the correct values and evaluate to find the answer.
When this lesson is done, you should be able to:
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Back To CourseGeometry: High School
15 chapters | 160 lessons