The prism: volume and surface area

The plane and space, grade 12 (3 units) Lesson 3 of 15Learning

The prism: volume and surface area

The prism

A prism has two bases that are equal, parallel polygons, and the faces between them are rectangles. A box is a prism whose base is a rectangle.

A prism is named after its base: a triangular prism, a pentagonal prism…
The height of a prism, $\displaystyle h$ , is the distance between its two bases – the length of the rectangular faces.
A prism can also lie on a rectangular face: then its bases are its front and back, and its height is its length. So it is with the tent in the drawing: its bases are the triangles.

The volume of a prism

Just as in a box: layers of cubes lie on the base, one on top of the other, all along the height. Each layer holds as many unit cubes as the area of the base, $\displaystyle B$ :

\displaystyle V

\displaystyle =

\displaystyle B\cdot h

The surface area of a prism

The lateral surface of a prism is its rectangular faces. Laid out side by side, they form one rectangle: its width is the perimeter of the base, $\displaystyle P$ , and its length is the height, $\displaystyle h$ .
The surface area is the two bases plus the lateral surface.

\displaystyle L

\displaystyle =

\displaystyle P\cdot h

\displaystyle A

\displaystyle =

\displaystyle 2B+L

Example: a tent

The tent in the drawing: at the front an isosceles triangle with a base of 3.2 m and a height of 1.2 m. Each of its two sloping sides is 2 m.
The tent is 2.5 m long, and it has a floor.
What is the tent’s volume, and how many m² of cloth does it have?

The base: $\displaystyle {B}={\frac{3.2\cdot1.2}{2}}={1.92}$ m².
The volume: $\displaystyle {V}={1.92\cdot2.5}={4.8}$ m³.
The perimeter of the base: $\displaystyle {P}={2+2+3.2}={7.2}$ m, and the lateral surface – the roof on both sides and the floor: $\displaystyle {L}={7.2\cdot2.5}={18}$ m².
The cloth: $\displaystyle {A}={2\cdot1.92+18}={21.84}$ m².

Exercise 1

A pool is 20 m long and 10 m wide.
Its floor slopes: the depth is 1 m at the shallow end and 2 m at the deep end.
The pool is a prism lying on its side: its base is the wall in the drawing, a trapezium.

The drawing is not to scale: the depth is enlarged, so that the slope shows.

a

The area of the wall (the trapezium), in m²

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

The pool’s volume, in m³

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

The cost of the water, at 8 ₪ per m³

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

A wheelchair ramp is cast in concrete. It is a prism whose base is a right-angled triangle: the leg on the ground is 3 m, and the upright leg 0.5 m.
The ramp is 1.5 m wide.
A cubic metre of concrete costs 400 ₪.

a

How many m³ of concrete are needed?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

How much will the concrete cost?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

A chocolate bar’s package is a triangular prism.
Its base is an equilateral triangle: the side is 4 cm, and the triangle’s height is 3.5 cm (roughly).
The package is 20 cm long.
How many cm² of cardboard does the package have?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

A garden shed is a prism whose base is a pentagon: a rectangle of 3 m by 2 m, and above it a triangle 1 m high – the roof.
The shed is 4 m deep.
What is the shed’s volume, in m³?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: the plane and space, grade 12 (3 units)

The box and the cube

The volume of a box – $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ its three dimensions
$\displaystyle V$ $\displaystyle =$ $\displaystyle abc$
The surface area of a box – three pairs of equal faces
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2(ab+ac+bc)$
The volume of a cube – $\displaystyle a$ the edge
$\displaystyle V$ $\displaystyle =$ $\displaystyle a^3$
The surface area of a cube – six square faces
$\displaystyle A$ $\displaystyle =$ $\displaystyle 6a^2$

The prism

The volume of a prism – $\displaystyle B$ the base area, $\displaystyle h$ the height (the distance between the bases)
$\displaystyle V$ $\displaystyle =$ $\displaystyle B\cdot h$
The lateral area – $\displaystyle P$ the perimeter of the base
$\displaystyle L$ $\displaystyle =$ $\displaystyle P\cdot h$
The surface area – the two bases and the lateral surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2B+L$

The cylinder

The volume of a cylinder – $\displaystyle r$ the radius of the base, $\displaystyle h$ the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \pi r^2h$
The curved surface area – a rectangle as long as the base’s circumference
$\displaystyle L$ $\displaystyle =$ $\displaystyle 2\pi rh$
The surface area – the two bases and the curved surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2\pi r^2+2\pi rh$

The pyramid and the cone

The volume of a pyramid – a third of the prism with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{B\cdot h}{3}$
In a square pyramid – $\displaystyle k$ the height of a side face, $\displaystyle h$ the pyramid’s height, $\displaystyle a$ the side of the base
$\displaystyle k^2$ $\displaystyle =$ $\displaystyle h^2+\left(\frac{a}{2}\right)^2$
The surface area of a square pyramid – the base and four triangles
$\displaystyle A$ $\displaystyle =$ $\displaystyle a^2+2ak$
The volume of a cone – a third of the cylinder with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{\pi r^2h}{3}$
The slant height $\displaystyle l$ – the hypotenuse of a triangle whose legs are the radius $\displaystyle r$ and the height $\displaystyle h$
$\displaystyle l^2$ $\displaystyle =$ $\displaystyle r^2+h^2$
The curved surface area of a cone
$\displaystyle L$ $\displaystyle =$ $\displaystyle \pi rl$
The surface area of a cone – the base and the curved surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2+\pi rl$

The sphere

The volume of a sphere – $\displaystyle r$ the radius
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{4\pi r^3}{3}$
The surface area of a sphere
$\displaystyle A$ $\displaystyle =$ $\displaystyle 4\pi r^2$

Diagonals and distances

The diagonal of a box $\displaystyle d$ – $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ its three dimensions
$\displaystyle d^2$ $\displaystyle =$ $\displaystyle a^2+b^2+c^2$
The diagonal of a cube – $\displaystyle a$ the edge
$\displaystyle d$ $\displaystyle =$ $\displaystyle a\sqrt{3}$
In a square pyramid – $\displaystyle s$ a side edge, $\displaystyle h$ the height, $\displaystyle a$ the side of the base
$\displaystyle s^2$ $\displaystyle =$ $\displaystyle h^2+\frac{a^2}{2}$

Trigonometry in a right-angled triangle

$\displaystyle a$ the leg opposite the angle $\displaystyle \alpha$ , $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{c}$
$\displaystyle b$ the leg next to the angle $\displaystyle \alpha$
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{b}{c}$
The leg opposite the angle over the leg next to it
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
The angle, when both legs are known
$\displaystyle \alpha=\tan^{-1}\left(\frac{a}{b}\right)$

The shortest path on the surface of a box

A path from a vertex to the vertex farthest from it, across two faces: unfolded, the diagonal of a rectangle with sides $\displaystyle a+b$ and $\displaystyle c$
$\displaystyle p^2$ $\displaystyle =$ $\displaystyle (a+b)^2+c^2$