Question
Find the surface area of the square pyramid. In order to receive full credit, you must show your work and label your answer with the appropriate units.
An illustration shows a 3 D pyramid with a square base. The side of the base is labeled 12 millimeters. The perpendicular height denoted by a right angle symbol on the lateral face from the apex to the base is labeled 10 millimeters.
An illustration shows a 3 D pyramid with a square base. The side of the base is labeled 12 millimeters. The perpendicular height denoted by a right angle symbol on the lateral face from the apex to the base is labeled 10 millimeters.
Answers
GPT-4o mini
To find the surface area of a square pyramid, we need to calculate the area of the base and the areas of the four triangular faces.
**Step 1: Calculate the area of the base.**
The base of the pyramid is a square. The area \( A_{base} \) of a square is given by the formula:
\[
A_{base} = s^2
\]
where \( s \) is the length of a side of the square.
In this case, the side of the base is 12 millimeters:
\[
A_{base} = 12^2 = 144 \text{ mm}^2
\]
**Step 2: Calculate the area of one triangular face.**
The area \( A_{triangle} \) of a triangle is given by the formula:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the triangular face of the pyramid, the base is equal to the side length of the square base (12 mm), and the height is the slant height of the pyramid.
However, we are given the perpendicular height from the apex to the base of the pyramid (10 mm). To find the slant height (\( l \)), we can use the Pythagorean theorem. The lateral face forms a right triangle where:
- one leg is the height of the pyramid (10 mm),
- the other leg is half the length of the base (6 mm, since \( \frac{12 \text{ mm}}{2} = 6 \text{ mm} \)),
- and the hypotenuse is the slant height (\( l \)).
Using the Pythagorean theorem:
\[
l = \sqrt{(height)^2 + \left(\frac{s}{2}\right)^2} = \sqrt{10^2 + 6^2} = \sqrt{100 + 36} = \sqrt{136} = 2\sqrt{34} \text{ mm}
\]
**Now calculating the area of one triangular face:**
\[
A_{triangle} = \frac{1}{2} \times 12 \times 2\sqrt{34}
\]
\[
A_{triangle} = 12\sqrt{34} \text{ mm}^2
\]
**Step 3: Calculate the total area of the four triangular faces.**
Since there are four triangular faces, the total area of the triangular faces \( A_{triangles} \) is:
\[
A_{triangles} = 4 \times A_{triangle} = 4 \times 12\sqrt{34}
\]
\[
A_{triangles} = 48\sqrt{34} \text{ mm}^2
\]
**Step 4: Calculate the total surface area of the pyramid.**
The total surface area \( A_{total} \) of the square pyramid is the sum of the area of the base and the area of the triangular faces:
\[
A_{total} = A_{base} + A_{triangles}
\]
\[
A_{total} = 144 + 48\sqrt{34} \text{ mm}^2
\]
Therefore, the surface area of the square pyramid is:
\[
\boxed{144 + 48\sqrt{34} \text{ mm}^2}
\]
**Step 1: Calculate the area of the base.**
The base of the pyramid is a square. The area \( A_{base} \) of a square is given by the formula:
\[
A_{base} = s^2
\]
where \( s \) is the length of a side of the square.
In this case, the side of the base is 12 millimeters:
\[
A_{base} = 12^2 = 144 \text{ mm}^2
\]
**Step 2: Calculate the area of one triangular face.**
The area \( A_{triangle} \) of a triangle is given by the formula:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the triangular face of the pyramid, the base is equal to the side length of the square base (12 mm), and the height is the slant height of the pyramid.
However, we are given the perpendicular height from the apex to the base of the pyramid (10 mm). To find the slant height (\( l \)), we can use the Pythagorean theorem. The lateral face forms a right triangle where:
- one leg is the height of the pyramid (10 mm),
- the other leg is half the length of the base (6 mm, since \( \frac{12 \text{ mm}}{2} = 6 \text{ mm} \)),
- and the hypotenuse is the slant height (\( l \)).
Using the Pythagorean theorem:
\[
l = \sqrt{(height)^2 + \left(\frac{s}{2}\right)^2} = \sqrt{10^2 + 6^2} = \sqrt{100 + 36} = \sqrt{136} = 2\sqrt{34} \text{ mm}
\]
**Now calculating the area of one triangular face:**
\[
A_{triangle} = \frac{1}{2} \times 12 \times 2\sqrt{34}
\]
\[
A_{triangle} = 12\sqrt{34} \text{ mm}^2
\]
**Step 3: Calculate the total area of the four triangular faces.**
Since there are four triangular faces, the total area of the triangular faces \( A_{triangles} \) is:
\[
A_{triangles} = 4 \times A_{triangle} = 4 \times 12\sqrt{34}
\]
\[
A_{triangles} = 48\sqrt{34} \text{ mm}^2
\]
**Step 4: Calculate the total surface area of the pyramid.**
The total surface area \( A_{total} \) of the square pyramid is the sum of the area of the base and the area of the triangular faces:
\[
A_{total} = A_{base} + A_{triangles}
\]
\[
A_{total} = 144 + 48\sqrt{34} \text{ mm}^2
\]
Therefore, the surface area of the square pyramid is:
\[
\boxed{144 + 48\sqrt{34} \text{ mm}^2}
\]