Joshua Rwothomio
5 hours ago
A rectangular pyramid with a triangular base has a slant length of 12 cm, given that each side of the base is 10 cm long. Find the surface area of the pyramid.
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ChatGPT
5 hours ago
To find the surface area of the pyramid, we need to find the area of the base and the area of the three triangular faces.
1. Area of the base:
Since the base is a triangle, we can find its area using the formula for the area of a triangle:
Area = 1/2 * base * height
In this case, the base of the triangle is 10 cm and the height can be found using the Pythagorean theorem:
height = sqrt(12^2 - 5^2) = sqrt(144 - 25) = sqrt(119) ≈ 10.91 cm
Therefore, the area of the base is:
Area_base = 1/2 * 10 * 10.91 = 54.55 cm^2
2. Area of the three triangular faces:
Each triangular face is an isosceles triangle with base 10 cm and slant height 12 cm. We can find the height of the triangle using the Pythagorean theorem:
height = sqrt(12^2 - 5^2) = sqrt(144 - 25) = sqrt(119) ≈ 10.91 cm
Therefore, the area of each triangular face is:
Area_face = 1/2 * base * height = 1/2 * 10 * 10.91 = 54.55 cm^2
Since there are three triangular faces, the total area of the three faces is:
Total_area_faces = 3 * 54.55 = 163.64 cm^2
3. Total surface area:
The total surface area of the pyramid is the sum of the area of the base and the area of the three faces:
Surface_area = Area_base + Total_area_faces
Surface_area = 54.55 + 163.64 = 218.18 cm^2
Therefore, the surface area of the rectangular pyramid is 218.18 cm^2.
1. Area of the base:
Since the base is a triangle, we can find its area using the formula for the area of a triangle:
Area = 1/2 * base * height
In this case, the base of the triangle is 10 cm and the height can be found using the Pythagorean theorem:
height = sqrt(12^2 - 5^2) = sqrt(144 - 25) = sqrt(119) ≈ 10.91 cm
Therefore, the area of the base is:
Area_base = 1/2 * 10 * 10.91 = 54.55 cm^2
2. Area of the three triangular faces:
Each triangular face is an isosceles triangle with base 10 cm and slant height 12 cm. We can find the height of the triangle using the Pythagorean theorem:
height = sqrt(12^2 - 5^2) = sqrt(144 - 25) = sqrt(119) ≈ 10.91 cm
Therefore, the area of each triangular face is:
Area_face = 1/2 * base * height = 1/2 * 10 * 10.91 = 54.55 cm^2
Since there are three triangular faces, the total area of the three faces is:
Total_area_faces = 3 * 54.55 = 163.64 cm^2
3. Total surface area:
The total surface area of the pyramid is the sum of the area of the base and the area of the three faces:
Surface_area = Area_base + Total_area_faces
Surface_area = 54.55 + 163.64 = 218.18 cm^2
Therefore, the surface area of the rectangular pyramid is 218.18 cm^2.
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Google Gemini AI
5 hours ago
The surface area of a rectangular pyramid with a triangular base is given by:
$$SA = \frac{1}{2} P\ell + B$$
where:
- P is the perimeter of the base
- \ell is the slant height
- B is the area of the base
Given:
- Each side of the base: 10 cm
- Slant height: 12 cm
First, find the perimeter of the base:
$$P = 3s = 3(10) = 30 cm$$
Next, find the area of the base:
$$B = \frac{1}{2} b h = \frac{1}{2} (10)(10) = 50 cm^2$$
Now, substitute these values into the formula for surface area:
$$SA = \frac{1}{2}(30)(12) + 50 = 180 + 50 = 230 cm^2$$
Therefore, the surface area of the rectangular pyramid is 230 cm².
$$SA = \frac{1}{2} P\ell + B$$
where:
- P is the perimeter of the base
- \ell is the slant height
- B is the area of the base
Given:
- Each side of the base: 10 cm
- Slant height: 12 cm
First, find the perimeter of the base:
$$P = 3s = 3(10) = 30 cm$$
Next, find the area of the base:
$$B = \frac{1}{2} b h = \frac{1}{2} (10)(10) = 50 cm^2$$
Now, substitute these values into the formula for surface area:
$$SA = \frac{1}{2}(30)(12) + 50 = 180 + 50 = 230 cm^2$$
Therefore, the surface area of the rectangular pyramid is 230 cm².
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