Let's break down the problem step by step.
Given:
- Length of an edge of the square base of the pyramid: 37 cm
- Length of the base of the pyramid: 24 cm
### Step 1: Find the Slant Height of the Pyramid
1. **Determine the height of each triangular face:**
The base of each triangle is one side of the square base of the pyramid (24 cm), and the edge of the pyramid is the hypotenuse of the right triangle formed by the slant height, the half base, and the perpendicular height of the triangle.
2. **Use the Pythagorean Theorem:**
The slant height (\( l \)) can be found using the Pythagorean theorem in the right triangle formed by half of the base side, the height of the pyramid, and the slant height. However, we need the perpendicular height of the pyramid first.
To find the perpendicular height (\( h \)):
- The half-base length: \( \frac{24}{2} = 12 \) cm
Applying the Pythagorean theorem in the triangle composed of the side edge (37 cm), half of the base (12 cm), and the height (\( h \)):
\[
h = \sqrt{37^2 - 12^2} = \sqrt{1369 - 144} = \sqrt{1225} = 35 \text{ cm}
\]
3. **Now find the slant height (\( l \)) using the height of the pyramid (35 cm) and the half base (12 cm):**
Applying the Pythagorean theorem:
\[
l = \sqrt{12^2 + 35^2} = \sqrt{144 + 1225} = \sqrt{1369} = 37 \text{ cm}
\]
Note: This shows that the edge length provided (37 cm) must be the slant height calculation, so the edge of the pyramid (375 cm) and the slant height (37 cm) are synonymous.
### Step 2: Find the Lateral Surface Area
The lateral surface area of the pyramid consists of 4 triangles.
1. **Calculate the area of one triangular face:**
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{Base} \times \text{Slant height}
\]
\[
= \frac{1}{2} \times 24 \text{ cm} \times 37 \text{ cm} = 12 \times 37 = 444 \text{ cm}^2
\]
2. **Total lateral surface area:**
\[
\text{Lateral Surface Area} = 4 \times \text{Area of one triangle} = 4 \times 444 = 1776 \text{ cm}^2
\]
### Step 3: Find the Total Surface Area
The total surface area of the pyramid is the sum of the lateral surface area and the area of the base.
1. **Area of the base:**
\[
\text{Area of the base} = 24 \text{ cm} \times 24 \text{ cm} = 576 \text{ cm}^2
\]
2. **Total Surface Area:**
\[
\text{Total Surface Area} = \text{Lateral Surface Area} + \text{Area of the base}
\]
\[
= 1776 \text{ cm}^2 + 576 \text{ cm}^2 = 2352 \text{ cm}^2
\]
### Summary:
1. **Slant height of the pyramid:** 37 cm
2. **Lateral surface area:** 1776 cm²
3. **Total surface area:** 2352 cm²
The length of edge of a square base pyramid is 37cm.The length of base is 24
1.Find the slant height of the pyramid
2.Find lateral surface area
3.Find the total surface area of the pyramid
1 answer