# The base of a triangular pyramid is a triangle with corners at (2 ,6 ), (5 ,3 ), and (8 ,2 ). If the pyramid has a height of 18 , what is the pyramid's volume?

Nov 26, 2017

Volume of the pyramid with triangular base is $18$

#### Explanation:

Volume of triangular pyramid = $\left(\frac{1}{3}\right) A H$ where A is the area of the triangular base and H is the height of the pyramid.

Area of triangular base = $\left(\frac{1}{2}\right) b h$ where b is the base and h is the height of the triangle.

Base $= \sqrt{{\left(5 - 2\right)}^{2} + {\left(3 - 6\right)}^{2}} = \sqrt{{3}^{2} + {3}^{2}} = \textcolor{b l u e}{3 \sqrt{2}}$

Eqn of Base =is
$\left(\frac{y - {y}_{1}}{{y}_{2} - {y}_{1}}\right) = \left(\frac{x - {x}_{1}}{{x}_{2} - {x}_{1}}\right)$

$\left(\frac{y - 6}{3 - 6}\right) = \left(\frac{x - 2}{5 - 2}\right)$

$\left(y - 6\right) = \left(- x + 2\right)$
$y + x = 8$ Eqn. (1)
Slope of base $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$
Slope $m = \frac{3 - 6}{5 - 2} = - 1$

Slope of altitude ${m}_{1} = - \left(\frac{1}{m}\right) = - \left(\frac{1}{- 1}\right) = 1$

Eqn of Altitude is
$\left(y - {y}_{3}\right) = {m}_{1} \left(x - {x}_{3}\right)$

$y - 2 = 1 \left(x - 8\right)$
$y - x = - 6$. Eqn (2)

Solving Eqns (1) & (2), we get coordinates of the base of the altitude.
Coordinates of base of altitude are (7,1)

Height of altitude$= \sqrt{{\left(8 - 7\right)}^{2} + {\left(2 - 1\right)}^{2}} = \textcolor{red}{\sqrt{2}}$

Area of triangular base $= \left(\frac{1}{2}\right) \textcolor{b l u e}{3 \sqrt{2}} \textcolor{red}{\sqrt{2}}$

Area $= 3$

Volume of pyramid = $\left(\frac{1}{3}\right) \cdot 3 \cdot 18 = 18$
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 18, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://socratic.org/questions/the-base-of-a-triangular-pyramid-is-a-triangle-with-corners-at-2-6-5-3-and-8-2-i-1", "fetch_time": 1716616018000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-22/segments/1715971058773.28/warc/CC-MAIN-20240525035213-20240525065213-00379.warc.gz", "warc_record_offset": 459399323, "warc_record_length": 6545, "token_count": 653, "char_count": 1644, "score": 4.71875, "int_score": 5, "crawl": "CC-MAIN-2024-22", "snapshot_type": "latest", "language": "en", "language_score": 0.5965399742126465, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00010-of-00064.parquet"}