Question

what is the surface area of this triangular pyramid? 16.4 mm 18 mm 16.4 mm 14.2 mm 16.4 mm square millimeters submit

Explanation:

Step1: Analyze the triangular pyramid

A triangular pyramid (tetrahedron) with a triangular base and three triangular faces. Here, the base is an equilateral triangle with side length \( 16.4 \) mm, and the height of the base triangle (altitude) is \( 14.2 \) mm. The three lateral faces are congruent triangles with base \( 16.4 \) mm and slant height \( 18 \) mm? Wait, no, the given slant height? Wait, the base area: area of the triangular base is \( \frac{1}{2} \times \text{base} \times \text{height of base} \). Then the lateral surface area: three times the area of one lateral triangle. Wait, looking at the diagram, the base triangle has sides \( 16.4 \) mm, \( 16.4 \) mm, \( 16.4 \) mm? Wait, no, the base is a triangle with base \( 16.4 \) mm and height \( 14.2 \) mm? Wait, no, the base is an equilateral triangle? Wait, the three edges at the base are \( 16.4 \) mm each? Wait, maybe the base is an equilateral triangle with side \( 16.4 \) mm, and the height of the base triangle (altitude) is \( 14.2 \) mm? Wait, no, let's re-express:

Wait, the surface area of a triangular pyramid (tetrahedron) is the sum of the area of the base and the area of the three lateral faces.

First, the base: it's a triangle with base \( b = 16.4 \) mm and height \( h_{base} = 14.2 \) mm. So area of base \( A_{base} = \frac{1}{2} \times 16.4 \times 14.2 \).

Then, the lateral faces: each lateral face is a triangle with base \( 16.4 \) mm and slant height (height of the lateral triangle) \( 18 \) mm? Wait, the diagram shows a slant height of \( 18 \) mm? Wait, no, the given lengths: the three lateral edges? Wait, no, the lateral faces: each has a base of \( 16.4 \) mm and a height (slant height) of \( 18 \) mm? Wait, maybe I misread. Wait, the triangular pyramid: the base is a triangle, and the three lateral faces are triangles. Let's check the given values:

Wait, the base triangle: base \( 16.4 \) mm, height \( 14.2 \) mm. So area of base: \( \frac{1}{2} \times 16.4 \times 14.2 \).

Then, the three lateral faces: each has a base of \( 16.4 \) mm and a height (slant height) of \( 18 \) mm? Wait, no, the diagram has a slant height of \( 18 \) mm? Wait, maybe the lateral faces have base \( 16.4 \) mm and height \( 18 \) mm? Wait, let's calculate:

First, base area: \( \frac{1}{2} \times 16.4 \times 14.2 \). Let's compute that: \( 0.5 \times 16.4 = 8.2 \); \( 8.2 \times 14.2 = 116.44 \) square mm.

Then, each lateral face: area is \( \frac{1}{2} \times 16.4 \times 18 \). There are three lateral faces, so lateral surface area \( A_{lateral} = 3 \times \frac{1}{2} \times 16.4 \times 18 \).

Wait, but wait, the diagram shows three edges of \( 16.4 \) mm at the base, and a slant height of \( 18 \) mm? Wait, maybe the base is an equilateral triangle? Wait, no, the height of the base is \( 14.2 \) mm, so it's an isoceles triangle? Wait, no, maybe the base is a triangle with sides \( 16.4 \) mm, \( 16.4 \) mm, and \( 16.4 \) mm? Wait, no, the height of the base is \( 14.2 \) mm, so if it's equilateral, the height would be \( \frac{\sqrt{3}}{2} \times 16.4 \approx 14.2 \) mm (since \( \frac{\sqrt{3}}{2} \times 16.4 \approx 0.866 \times 16.4 \approx 14.2 \)). Oh! So the base is an equilateral triangle with side length \( 16.4 \) mm, and its height is \( \frac{\sqrt{3}}{2} \times 16.4 \approx 14.2 \) mm, which matches the given \( 14.2 \) mm. So the base is equilateral with side \( 16.4 \) mm, height \( 14.2 \) mm.

Then, the lateral faces: each is a triangle with base \( 16.4 \) mm and slant height \( 18 \) mm? Wait, the diagram shows a slant height of \( 18 \) mm. So each lateral face has area \( \frac{1}{2} \times 16.4 \times 18 \), and there are three of them.

So total surface area \( A = A_{base} + A_{lateral} \).

Let's compute each part:

Step1: Calculate base area

Base is an equilateral triangle with side \( a = 16.4 \) mm and height \( h_{base} = 14.2 \) mm.
\( A_{base} = \frac{1}{2} \times a \times h_{base} = \frac{1}{2} \times 16.4 \times 14.2 \)
\( 16.4 \times 14.2 = 232.88 \)
\( \frac{1}{2} \times 232.88 = 116.44 \) square mm.

Step2: Calculate lateral surface area

Each lateral face is a triangle with base \( a = 16.4 \) mm and height (slant height) \( l = 18 \) mm. There are 3 lateral faces.
Area of one lateral face: \( \frac{1}{2} \times a \times l = \frac{1}{2} \times 16.4 \times 18 \)
\( 16.4 \times 18 = 295.2 \)
\( \frac{1}{2} \times 295.2 = 147.6 \) square mm per lateral face.
Total lateral surface area: \( 3 \times 147.6 = 442.8 \) square mm.

Step3: Calculate total surface area

Total surface area \( A = A_{base} + A_{lateral} = 116.44 + 442.8 = 559.24 \)? Wait, that can't be right. Wait, no, wait: maybe the lateral faces have a different height. Wait, maybe the slant height is \( 16.4 \)? No, the diagram shows \( 18 \) mm. Wait, maybe I made a mistake. Wait, the triangular pyramid: if the base is equilateral with side \( 16.4 \), and the three lateral faces are congruent triangles with base \( 16.4 \) and height \( 18 \), then:

Wait, let's recalculate:

Base area: \( \frac{1}{2} \times 16.4 \times 14.2 = 116.44 \) (correct).

Lateral face area: each is \( \frac{1}{2} \times 16.4 \times 18 = 147.6 \). Three of them: \( 3 \times 147.6 = 442.8 \).

Total surface area: \( 116.44 + 442.8 = 559.24 \). But wait, maybe the lateral faces have a height of \( 16.4 \)? No, the diagram shows \( 18 \) mm. Wait, maybe the given \( 18 \) mm is the slant height. Wait, maybe I misread the diagram. Alternatively, maybe the triangular pyramid is a regular tetrahedron? No, because the base height is \( 14.2 \), which is \( \frac{\sqrt{3}}{2} \times 16.4 \approx 14.2 \), so the base is equilateral, and the lateral faces: maybe the slant height is \( 18 \) mm. So the calculation seems correct. Wait, but let's check again:

Wait, \( \frac{1}{2} \times 16.4 \times 14.2 = 8.2 \times 14.2 = 116.44 \) (correct).

\( \frac{1}{2} \times 16.4 \times 18 = 8.2 \times 18 = 147.6 \) (correct). Three of them: \( 3 \times 147.6 = 442.8 \) (correct).

Total: \( 116.44 + 442.8 = 559.24 \). Wait, but maybe the lateral faces are not with slant height \( 18 \), but with height \( 16.4 \)? No, the diagram shows \( 18 \) mm. Wait, maybe the problem is that the triangular pyramid has a base that is an equilateral triangle, and the three lateral faces are isoceles triangles with base \( 16.4 \) and height \( 18 \). So the calculation is as above.

Wait, but let's check the numbers again. Maybe I made a mistake in the base area. Wait, the base is a triangle with base \( 16.4 \) and height \( 14.2 \), so area is \( 0.5 * 16.4 * 14.2 \). Let's compute \( 16.4 * 14.2 \):

\( 16 * 14.2 = 227.2 \)

\( 0.4 * 14.2 = 5.68 \)

Total \( 227.2 + 5.68 = 232.88 \)

Then \( 0.5 * 232.88 = 116.44 \) (correct).

Lateral faces: each is \( 0.5 * 16.4 * 18 \). \( 16.4 * 18 = 295.2 \), \( 0.5 * 295.2 = 147.6 \). Three of them: \( 3 * 147.6 = 442.8 \).

Total surface area: \( 116.44 + 442.8 = 559.24 \). So the surface area is \( 559.24 \) square millimeters.

Wait, but maybe the lateral faces have a different height. Wait, the diagram shows a length of \( 18 \) mm as the slant height. So I think this is correct.

Answer:

\( 559.24 \)