You are in History class learning about different artifacts from other cultures, when the subject of pyramids is presented by your teacher. He informs the class that pyramids come in a variety of sizes, designs, and styles, and are found not only in Egypt, but in many other countries around the world. He tells everyone that a typical pyramid might be approximately 200 meters long at the base and 175 meters up each of the diagonal sides.
Your mind wanders back to your math class from that morning, and you find yourself wondering if there is a straightforward way to determine the area of one of the faces of the pyramid from the information you have been given.
Do you think there is a way to do this?
As it turns out, there is a very straightforward way to determine the area of a triangle when you know the lengths of all three sides. At the end of this Concept, you'll be able to calculate the area of one of the pyramid's sides.
James Sousa: Heron's Formula
Solution: In order to find the cost of the tiles, we first need to find the area of one tile. Since we know the measurements of all three sides, we can use Heron’s Formula to calculate the area.
The area of one tile is 63.843 square inches. The cost of the tile is given to us in square feet, while the area of the tile is in square inches. In order to find the cost of one tile, we must first convert the area of the tile into square feet.
Heron's Formula: Heron's formula is a formula to calculate the area of a triangle when the lengths of all three sides are known.
2. The Pyramid Hotel is planning on repainting the exterior of the building. The building has four sides that are isosceles triangles with bases measuring 590 ft and legs measuring 375 ft.
a. What is the total area that needs to be painted? b. If one gallon of paint covers 25 square feet, how many gallons of paint are needed?
2. a. Use Heron’s Formula, then multiply your answer by 4, for the 4 sides.
Concept Problem Solution
You can use Heron's Formula to find the area of one of the faces of the pyramid.
so, in this case,
Find the area of each triangle with the three given side lengths.
- 2, 14, 15
- 6, 8, 9
- 10, 14, 20
- 11, 15, 6
- 4, 4, 4
- 4, 5, 3
- 32, 40, 50
- 20, 18, 22
- 20, 20, 20
- 18, 17, 12
- 9, 12, 10
- 11, 18, 8
- Describe when it makes the most sense to use Heron's formula to find the area of a triangle.
- A tiling is made of 30 congruent triangles. The lengths of the sides of each triangle are 3 inches, 5 inches, and 7 inches. What is the area of the tiling?
- What type of triangle with have the maximum area for a given perimeter? Show or explain your reasoning.