LemonmathSelf-study sheet

Grade 12 · Lesson 7.5

Revision: geometry in space

Name: Date: Time spent: min

Goals

Prove quickly that a line is perpendicular to a plane and recognise perpendicular planes.
Identify and calculate the angle between a line and a plane, and a dihedral angle.
Calculate the distance from a point to a plane and between two skew lines.
Calculate the volume of pyramids and prisms, including in real-life problems.

Exercises

Exercise 1 Multiple choice

The right prism $ABC.A'B'C'$ has a base that is a right-angled triangle at $A$ , with $AB=3$ , $AC=4$ and $AA'=5$ . The volume of the prism is

A. $30$ . B. $10$ . C. $60$ . D. $20$ .

Exercise 2 Multiple choice

In the tetrahedron $OABC$ , the edges $OA$ , $OB$ , $OC$ are mutually perpendicular, $OA=2$ , $OB=3$ , $OC=4$ . The volume of the tetrahedron is

A. $24$ . B. $12$ . C. $8$ . D. $4$ .

Exercise 3 Multiple choice

The pyramid $S.ABC$ has an equilateral base with side $a$ , $SA\perp(ABC)$ , and the angle between the plane $(SBC)$ and the base is $60^\circ$ . The volume of the pyramid is

A. $\dfrac{a^3\sqrt3}{24}$ . B. $\dfrac{a^3\sqrt3}{8}$ . C. $\dfrac{3a^3\sqrt3}{8}$ . D. $\dfrac{a^3}{4}$ .

Exercise 4 True/false

$ABCD.A'B'C'D'$ is a cube with side $a$ , and $O$ is the centre of the square $ABCD$ .

a) $AC\perp(BDD'B')$ .

b) The angle between the lines $AB'$ and $BC'$ is $60^\circ$ .

c) The distance from $A$ to the plane $(BDD'B')$ is $\dfrac{a\sqrt2}{2}$ .

d) The dihedral angle $[C', BD, C]$ is $45^\circ$ .

Exercise 5 Short answer

$ABCD.A'B'C'D'$ is a cube. Find the angle between the line $A'C$ and the plane $(ABCD)$ (in degrees, to the nearest whole number).

Exercise 6 Short answer

The pyramid $S.ABC$ has an equilateral base with side $2$ , $SA\perp(ABC)$ and $SA=1$ . Find the distance from $A$ to the plane $(SBC)$ (to 2 decimal places).

Exercise 7 Short answer

The cover of a skylight is a regular square pyramid with a square base of side $4$ m and height $3$ m. Find the angle between a sloping face of the cover and the base (in degrees, to the nearest whole number).

Exercise 8 Short answer

A water trough $2$ m long is a right prism whose cross-section is an isosceles trapezium with a longer base (the top) of $0.6$ m, a shorter base of $0.4$ m and a depth of $0.3$ m. At most how many litres of water can the trough hold?

Exercise 9 Written answer

The pyramid $S.ABCD$ has a square base with side $a$ and centre $O$ , $SA\perp(ABCD)$ and $SA=a$ . Find the distance between the skew lines $SC$ and $BD$ .

Exercise 10 Short answer (challenge)

The prism $ABC.A'B'C'$ has volume $30$ . Let $M$ be the midpoint of the edge $CC'$ . The pyramid $M.ABC$ is cut away from the prism. Find the volume of what remains.

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. A pyramid has base area

12

and height

5

. Its volume is

60

.B.

20

.C.

30

.D.

10

Question 2. The pyramid

S.ABCD

has

SA\perp(ABCD)

. The angle between the line

SB

and the plane

(ABCD)

\angle SBA

.B.

\angle BSA

.C.

\angle SBC

.D.

\angle SAB

Question 3.

ABCD.A'B'C'D'

is a cube with side

2

. The distance between the lines

AA'

and

BD

2

.B.

2\sqrt2

.C.

\sqrt2

.D.

\sqrt3

Question 4. The pyramid

S.ABCD

has a rectangular base and

SA\perp(ABCD)

. Which plane is perpendicular to the plane

(SAB)

(SCD)

.B.

(SBD)

.C.

(SAC)

.D.

(SBC)

Part II. True or false

Question 1.

S.ABCD

is a regular square pyramid with all edges equal to

2

, and

O

is the centre of the base.

a) $SO\perp(ABCD)$ .	T / F
b) $SO=\sqrt2$ .	T / F
c) The angle between $SA$ and the plane $(ABCD)$ is $60^\circ$ .	T / F
d) The volume of the pyramid is $\dfrac{4\sqrt2}{3}$ .	T / F

Part III. Short answer

Question 1. A right prism has an equilateral base with side

2

and lateral edges of length

3

. Find its volume (to 1 decimal place).

Question 2. The pyramid

S.ABC

has

SA\perp(ABC)

, triangle

ABC

has its right angle at

B

, and

SA=AB=2

. Find the distance from

A

to the plane

(SBC)

(to 2 decimal places).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $V_{\text{prism}}=B\cdot h$ ; in a right prism the height equals the lateral edge.

Steps:

Find the area of the right-angled triangle $ABC$ .
Multiply by $AA'$ .

Exercise 2.

What you need: $V=\dfrac16\,OA\cdot OB\cdot OC$ .

Steps:

Take $OAB$ (a right-angled triangle) as the base and $OC$ as the height.
Compute $\dfrac13\cdot S_{OAB}\cdot OC$ .

Exercise 3.

What you need: let $M$ be the midpoint of $BC$ ; since $BC\perp(SAM)$ , the angle between $(SBC)$ and the base is $\angle SMA$ .

Steps:

Find $AM=\dfrac{a\sqrt3}{2}$ , then $SA=AM\tan60^\circ$ .
Compute $V=\dfrac13\cdot\dfrac{a^2\sqrt3}{4}\cdot SA$ .

Exercise 4.

What you need: the criterion for a line perpendicular to a plane; the angle between two lines equals the angle between two lines parallel to them.

Steps:

$AC\perp BD$ and $AC\perp BB'$ .
Replace $BC'$ by the parallel line $AD'$ and look at triangle $AB'D'$ .
For the dihedral angle, use $C'O\perp BD$ and $CO\perp BD$ .

Exercise 5.

What you need: the projection of $A'C$ on $(ABCD)$ is $AC$ .

Steps:

Call the side $a$ and find $AC=a\sqrt2$ .
Compute $\tan\angle A'CA=\dfrac{AA'}{AC}$ .

Exercise 6.

What you need: with $M$ the midpoint of $BC$ , $BC\perp(SAM)$ ; $\dfrac{1}{AH^2}=\dfrac{1}{SA^2}+\dfrac{1}{AM^2}$ .

Steps:

Find $AM=\sqrt3$ .
Substitute into the formula.

Exercise 7.

What you need: with $O$ the centre of the base and $M$ the midpoint of a base edge, the angle between the face and the base is $\angle SMO$ .

Steps:

$OM$ is half the side of the base.
Compute $\tan\angle SMO=\dfrac{SO}{OM}$ .

Exercise 8.

What you need: $V=B\cdot h$ with a trapezium as the base; $1$ m³ $=1000$ litres.

Steps:

Find the area of the trapezium.
Multiply by the length of the trough, then convert to litres.

Exercise 9.

What you need: $BD\perp(SAC)$ ; if $OH\perp SC$ in the plane $(SAC)$ , then $OH$ is the common perpendicular.

Steps:

Show that $BD\perp OH$ .
Compute $OH=OC\cdot\sin\angle SCA$ .

Exercise 10.

What you need: the pyramid and the prism share the base; the height of the pyramid is half the height of the prism.

Steps:

Write $V_{M.ABC}=\dfrac13\cdot B\cdot\dfrac h2$ in terms of $V=B\cdot h$ .
Subtract it from $30$ .

Self-check

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

Redo the ones you missed after: ☐ 1 day ☐ 3 days ☐ 7 days

Scan for step-by-step solutions

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐